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On Post Critically Finite Polynomials

arXiv:math/9307235v1 [math.DS] 10 Jul 1993On Post Critically Finite PolynomialsPart Two: Hubbard TreesAlfredo PoirierMath DepartmentSuny@StonyBrookStonyBrook, NY 11794-3651We provide an effective classification of postcritically finite polynomials as dynamicalsystems by means of Hubbard Trees.This paper is the second in a series of two based on the author’s thesis which dealswith the classification of postcritically finite polynomials as dynamical systems (see [P2]).In the first part [P1], we conclude the study of critical portraits initiated by Fisher (see[F]) and continued by Bielefeld, Fisher and Hubbard (see [BHF]). As an application of ourresults, we give in this second part of the series necessary and sufficient conditions for therealization of Hubbard Trees.1

Hubbard Trees.Given a polynomial P of degree n ≥2, we consider the set K(P) (called the filled Juliaset) of points whose orbit under iteration is bounded. This set is known to be compact andits complement consists of a unique unbounded component (see [M, Lemma 17.1]).

Thebehavior under iteration of the critical points of this polynomial dramatically influences thetopology of this set K(P). For example, this set is connected if and only if all critical pointsare contained within (see [M, Theorem 17.3]).

We are interested in the special case wherethe orbit of every critical points is finite, i.e, the case where the orbits of all critical pointsare periodic or eventually periodic. We call such polynomials postcritically finite (PCF inshort).

For such polynomials the filled Julia set K(P) is connected. Furthermore, it is alsoknown in this case that K(P) is locally connected (see [M, Theorem 17.5]).Inordertoproceedfurtherweestablishsomenotation.ThesetJ(P) = ∂K(P) is called the Julia set, and its elements Julia points.The complementF(P) = C −J(P) of the Julia set is called the Fatou set and its elements Fatou points.

Aperiodic orbit z0 7→z1 = P(z0) 7→. .

. 7→zn = z0 which contains a critical point is called acritical cycle.

In the PCF case a periodic orbit belongs to the Fatou set F(P) if and onlyif it is a critical cycle (see [M, Corollary 11.6]).In this PCF case the dynamics of the polynomial admits a further decomposition.When restricted to the interior of K(P) (which is not empty if and only if there existsa critical cycle), P maps each component onto some other as a branched covering map.Furthermore, every component is eventually periodic (see [M, Theorem 13.4]). It is wellknown (see [M, Theorem 6.7]) that each component can be uniformized so that in localcoordinates P can be written as z 7→zn for some n ≥1.

Furthermore, if U is a periodicbounded Fatou component, the first return map is conjugate in local coordinates to z 7→znfor some n ≥2. In particular such cycles of components are in one to one correspondencewith critical cycles.

Also, in each component there is a unique point which eventually mapsto a critical point (these points are those which correspond to 0 in local coordinates).In the work [DH1], Douady and Hubbard suggested a combinatorial description of thedynamics of such polynomials using a tree-like structure. First we note the following (see[DH1, Corollary VII.4.2 p 64]).Lemma.

Let P be a PCF polynomial. Then for any z ∈J(P), the set J(P) −{z}consists of only a finite number of connected components.2

Thus, the filled Julia set is arranged in a tree like fashion. To simplify this tree weconsider a finite invariant set M (i.e, P(M) ⊂M) containing all critical points.

We jointhem in K(P) by paths subject to the restriction that if they intersect a Fatou component,this intersection must consist of radial segments in the coordinate described above. Douadyand Hubbard proved that this construction is unique and defines a finite topological tree TMin which all points in M (and perhaps more) are vertices.

Now, if from this tree we retainthe dynamics and local degree at every vertex, the way this tree is embedded in the complexplane (up to isotopy class), and “a bit of extra information to recover the tree generatedby P −1(M)” (there are several ways to state this condition in a non-ambiguous way), theyproved that different PCF polynomials (i.e, not conjugated as dynamical systems) give riseto different tree-structures. No criterion for realization was given at the time.

(The onlyprevious partial results about realization are given in Lavaurs’ thesis [L]).A way to deal with this conditions is to introduce angles around vertices in the treestructure (see [DH1, p.46]). In what follows we will measure angles in turns (i.e, 360◦= 1turn).

Around a Fatou vertex v (which correspond to 0 in the uniformizing coordinate),an angle between edges incident at v is naturally defined by means of the local coordinatesystem. At Julia vertices, where m components of K(P) meet (compare the lemma above),the angle is defined to be a multiple of 1/m (this normalization is introduced here for the firsttime).

These angles satisfy two conditions. First, they are compatible with the embeddingof the tree.

Second, we have that ̸P (v)(P(ℓ), P(ℓ′)) = δ(v)̸v(ℓ, ℓ′) (mod 1), where δ(v) isthe local degree of P at v and ℓ, ℓ′ are edges incident at v (̸v and ̸P (v) measure the anglesat v and P(v) respectively). When this further structure is given, we have a ‘dynamicaltree’, which we denote by HP,M.Now let us start with an abstract tree and try to reconstruct the appropriate polynomial.Definition.

By an (angled) tree H will be meant a finite connected acyclic m-dimensionalsimplicial complex (m = 0, 1), together with a function ℓ, ℓ′ 7↛ (ℓ, ℓ′) = ̸v(ℓ, ℓ′) ∈Q/Zwhich assigns a rational modulo 1 to each pair of edges ℓ, ℓ′ which meet at a common vertexv. This angle ̸ (ℓ, ℓ′) should be skew-symmetric, with ̸ (ℓ, ℓ′) = 0 if and only if ℓ= ℓ′, andwith ̸v(ℓ, ℓ′′) = ̸v(ℓ, ℓ′) + ̸v(ℓ′, ℓ′′) whenever three edges are incident at a vertex v. Suchan angle function determines a preferred isotopy class of embeddings of H into C.Let V be the set of vertices.

We specify a mapping τ : V →V and call it the vertexdynamics, and require that τ(v) ̸= τ(v′) whenever v and v′ are endpoints of a common edgeℓ. We consider also a local degree function δ : V →Z which assigns an integer δ(v) ≥1to each vertex v ∈V .

We require that deg(δ) = 1 + Pv∈V (δ(v) −1) be greater that 1.By definition a vertex v is critical if δ(v) > 1, and non-critical otherwise. The critical setΩ(δ) = {v ∈V : v is critical} is thus non empty.The maps τ and δ must be related in the following way.

Extend τ to a map τ : H →Hwhich carries each edge homeomorphically onto the shortest path joining the images of its3

endpoints. We require then that ̸τ(v)(τ(ℓ), τ(ℓ′)) = δ(v)̸v(ℓ, ℓ′) whenever ℓ, ℓ′ are incidentat v (in this case τ(ℓ) and τ(ℓ′) are incident at the vertex τ(v) where the angle is measured).A vertex v is periodic if for some n > 0, τ ◦n(v) = v. The orbit of a periodic criticalpoint is a critical cycle.

We say that a vertex v is of Fatou type or a Fatou vertex if iteventually maps into a critical cycle. Otherwise, if it eventually maps to a non critical cycle,it is of Julia type or a Julia vertex.We define the distance dH(v, v′) between vertices in H as the number of edges in ashortest path γ between v and v′.

We say that (H, V, τ, δ) is expanding if the followingcondition is satisfied. For any edge ℓwhose end points v, v′ are Julia vertices, there is ann ≥1 such that dH(τ ◦n(v), τ◦n(v′)) > 1.The angles at Julia vertices are rather artificial, so we normalize them as follows.

If medges ℓ1, . .

. , ℓm meet at a periodic Julia vertex v, then we assume that the angles ̸v(ℓi, ℓj)are all multiples of 1/m.

(It follows that the angles at a periodic Julia vertex convey noinformation beyond the cyclic order of these m incident edges. )By an abstract Hubbard Tree we mean an angled tree H = ((H, V, τ, δ), ̸ ) so that theangles at any periodic Julia vertex where m edges meet are multiples of 1/m.The basic existence and uniqueness theorem can now be stated as follows (compareTheorem II.4.7).Theorem A.

Any abstract Hubbard Tree H can be realized as a tree associated witha postcritically finite polynomial P if and only if H is expanding.Such a realization isnecessarily unique up to affine conjugation.This abstract Hubbard Tree also gives information about external rays as the followingtheoremessentiallyduetoDouadyandHubbardshows(compare [DH1, Chap VII]). This will follow in our case from Propositions II.3.3, III.4.3and the fact that J(P) is locally connected.Theorem B.

The number of rays which land at a periodic Julia vertex v is equal tothe number of incident edges of the tree T at v, and in fact, there is exactly one ray landingbetween each pair of consecutive edges. Furthermore, the ray which lands at v between ℓandℓ′ maps to the ray which lands at f(v) between f(ℓ) and f(ℓ′).After these theorems there is no reason to distinguish between the abstract HubbardTree and the unique polynomial which realizes it.4

Definition. A point p ∈J(P) is terminal if there is only one external ray landing atp.

Otherwise p is an incidence point. For incidence points we distinguish between branching(if there are more than two rays landing at p) and non branching (exactly two rays landingat p).

For a postcritically finite polynomial P, every branching point must be periodic orpreperiodic. Also every periodic branching point is present as vertex in any tree HP,M.Proposition I.3.2.Let P be a Postcritically Finite Polynomial and z ∈J(P) abranching point.

Then z is preperiodic (or periodic).Proposition I.3.3. Let P be a Postcritically Finite Polynomial and z ∈J(P) a periodicincidence point.

For any invariant finite set M containing the critical points of P, we havez ∈TP,M. Furthermore, the number of components of TP,M −{z} is independent of M andequals the number of components of J(P) −{z}.Now we give a brief description of the following chapters.

In Chapter I we have includedthe basic background of Hubbard Trees following the original exposition of Douady andHubbard. We have done so because there is nowhere in the literature where we can find in asystematic way what was known up to now.

In Chapter II, we introduce our basic abstractframework. We have carefully justified why there is the need to introduce all the abstractelements in our definition.

In Chapter III we give the proof of our main result. This proofis based in the theory of critical portraits developed in the first part of this work.

For theconvenience of the reader we have included in Appendix A an outline of this theory. InAppendix B, we study necessary and sufficient conditions under which an nth fold coveringof a finite cyclic set to a proper subset can be given a compatible ‘argument coordinate’ sothat it becomes multiplication by n.Acknowledgement.

We will like to thank John Milnor for helpful discussions andsuggestions. Some of the arguments are in its final formulation thanks to him.

Also, wewant to thank the Geometry Center, University of Minnesota and Universidad Cat´olica delPer´u for their material support.5

Chapter IHubbard TreesIn this Chapter we recall the definition and survey the main properties of Hubbard Treesas defined by Douady and Hubbard in [DH1]. In Section 1 we define the main concepts anddeduce some properties.

We ask the reader to pay special attention to Proposition 1.21.In Section 2 we define the inverse of Hubbard Trees. In Section 3 we define and study theincidence number at every point p of the tree and relate this concept with the number ofconnected components of J(P) −{p}.1.

Regulated Trees.1.1 Let P be a Postcritically Finite Polynomial. Given two points in the closure of abounded Fatou component, they can be joined in a unique way by a Jordan arc consisting of(at most two) segments of internal rays.

We call such arcs (following Douady and Hubbard)regulated. The filled Julia set K(P) being connected and locally connected in a compactmetric space is also arcwise connected.

This means that given two points z1, z2 ∈K(P)there is a continuous injective map γ : I = [0, 1] 7→K(P) such that γ(0) = z1 and γ(1) = z2.In general we will not distinguish between the map and its image. Such arcs (actually theirimages) can be chosen in a unique way so that the intersection with the closure of a Fatoucomponent is regulated (see [DH1, Chapter 2]).

We still call such arcs regulated, and denotethem by [z1, z2]K.The following immediate properties hold for regulated arcs (compare also [DH1, Chapter2]).1.2 Lemma. Let γ1, γ2 be regulated arcs, then γ1 ∩γ2 is regulated.#1.3 Lemma.

Every subarc of a regulated arc is regulated.#1.4 Lemma. Let z1, z2, z3 ∈K(P), then there exists p ∈K(P) such that [z1, z2]K ∩[z2, z3]K = [p, z2]K. In particular if [z1, z2]K ∩[z2, z3]K = {z2}, the set [z1, z2]K ∪[z2, z3]Kis a regulated arc.#6

1.5 Regulated Sets. We say that a subset X ⊂K(P) is regulated connected if forevery z1, z2 ∈X we have [z1, z2]K ⊂X.

We define the regulated hull [X]K of X ⊂K(P) asthe minimal closed regulated connected subset of K(P) containing X.1.6 Proposition. If z1, ..., zn are points in K(P), the regulated hull [z1, ..., zn]K of{z1, ..., zn} is a finite topological tree.Proof (Compare [DH1]).

The proof is by induction in the number of points. This isclearly true for small n (= 1, 2).

Suppose [z1, ..., zn]K is a finite topological tree, and letzn+1 ∈K(P). Let p any point in [z1, ..., zn]K and y the first point in the arc [zn+1, p]Kthat belongs to [z1, ..., zn]K.In this way [z1, ..., zn+1]K = [z1, ..., zn]K ∪[y, zn+1]K and[z1, ..., zn]K ∩[y, zn+1]K = {y}.

The result follows.#1.7 Remark. By definition every end of the tree [z1, ..., zn]K is one of zk, but theremay be zk which are not ends.1.8 Lemma.

Let γ(I) ⊂K(P) be a regulated arc containing no critical point of P,except possibly for its end points. Then P|γ(I) is injective and P(γ(I)) is a regulated arc.Proof.

The second part follows from the first, so let us show that P|γ is injective.As P ◦γ is locally one to one, the set ∆= {(t1, t2) : t1 < t2 and P(γ(t1)) = P(γ(t2))}is compact. If this set is non empty we can take (t1, t2) ∈∆with t2 −t1 minimal.

Lett ∈(t1, t2), then P(γ([t1, t])) and P(γ([t, t2])) are regulated arcs with the same end points;therefore they are equal and t2 −t1 is not minimal.#1.9 Definition. For a finite invariant set M, containing the set Ω(P) of critical pointsof P, we denote by T (M) the tree generated by M, i.e, the regulated hull [M]K.Theminimal tree T (M0), is the tree generated by M0 = O(Ω(P)) the orbit of the critical set.This last tree is usually called in the literature the Hubbard Tree of P.1.10 Lemma.

For a finite invariant set M, containing the set Ω(P) of all criticalpoints, P(T (M)) = [P(M)]K.Proof.The tree T (M) is the union of regulated arcs of the form [z1, z2]K withz1, z2 ∈M not containing a critical point except possibly for their end points. By Lemma1.8, P(T (M)) is the union of the regulated arcs [P(z1), P(z2)]K. As this set is regulatedconnected and contains all of P(M), by definition this set equals [P(M)]K.#1.11 Remark.

If X ⊂K(P) is arbitrary, the same argument shows that P(T (X)) ⊂[P(X ∪Ω(P))]K.7

1.12 Definition. Let T ∗(M) be the family whose elements are the closures of compo-nents of T (M) −Ω(P).1.13 Lemma.

P induces a continuous map from T (M) to itself, where the restrictionto every element (component) of T ∗(M) is injective.Proof. This follows from Lemmas 1.8 and 1.10.#1.14 Lemma.

Let γ(I) ⊂K(P) be a regulated arc containing no critical value of Pexcept possibly for its end points. Then any lift of γ(I) by P is a regulated arc.Proof.As γ|(0,1) contains no critical value of P, it can be pulled back by P in ddifferent ways, each being a regulated arc.#1.15 Definition.

Given z ∈T (M) the incidence number νT (M)(z) of T (M) at z is thenumber of components of T (M) −{z}. In other words, νT (M)(z) is the number of branchesof T (M) that are incident at z.

Note that this number might be different from the numberof connected components of K(P) −{z} (the incidence number at z for P).A point z ∈T (M) is called a branching point of T (M) if νT (M)(z) > 2 and an end ifνT (M)(z)=1.ThepreferredsetofT (M)isVT (M)=M∪{z ∈T (M) : νT (M)(z) > 2}. Note that VT (M) is finite.

This because there are only afinite number of vertices in this tree.1.16 Proposition. The set VT (M) is invariant.

Furthermore, it generates the sametree as M; i.e, T (M) = T (VT (M)).Proof. If z is a branching point and degzP = 1, then P(z) is also a branching point withνT (M)(P(z)) ≥νT (M)(z) because P maps T (M) into itself and P is a local homeomorphismin a neighborhood of z.We must prove that [M]K = [VT (M)]K. As M ⊂VT (M) then [M]K ⊂[VT (M)]K. Alsoby definition VT (M) ⊂[M]K, so [VT (M)]K ⊂[[M]K]K = [M]K.#1.17 Corollary.

Let M, M ′ be finite invariant subsets containing Ω(P). If VT (M) =VT (M′) then T (M) = T (M ′).#1.18 Proposition.

Let v, v′ ∈J(P) be two periodic points. If for all n ≥0, P ◦n(z)and P ◦n(z′) belong to the same element (component) of T ∗(M), then v = v′.8

Proof. Suppose P ◦n(v), P ◦n(v′) belong to the same component of T ∗(M) for all n ≥0.By Lemma 1.8 there is no precritical point in [v, v′]K. It follows easily that [v, v′]K ⊂J(P).Next, let m be the least common multiple of the periods of v and v′.

Thus, v, v′ are fixedby P ◦m. As there are only a finite number of such fixed points, we may assume that thereare no other in this set [v, v′]K. Both endpoints of this regulated arc are repelling.

Alsoby Lemma 1.8, P ◦m induces an homeomorphism of [v, v′]K onto itself. It follows that theremust be other fixed point in the interior of the arc [v, v′]K, in contradiction to what wasassumed.#1.19 Remark.

Note that the same is true if v, v′ are assumed only to be preperiodic.In this case, high enough iterates of both points must be periodic and therefore coincide.Lemma 1.13 will imply that v, v′ are identified as well.1.20 Definition. We define the distance dT (M)(v, v′) between points v, v′ ∈VT (M)as follows.Set dT (M)(v, v) = 0.Otherwise, take a regulated arc [v, v′]K and definedT (M)(v, v′) = #(VT (M) ∩[v, v′]K) −1 (# denotes as usual cardinality).Thus, dT (M)measures the number of ‘edges’ between v and v′.

In this language Proposition 1.18 can beread as follows.1.21 Proposition: Expanding Property of the tree T(M). For all pairs v, v′ ∈VT (M)∩J(P) satisfying dT (M)(v, v′) = 1, there is an n ≥1 such that dT (M)(P ◦n(v), P ◦n(v′)) >1.Proof.

As v, v′ are eventually periodic, the result follows from Proposition 1.18.#2. The Regulated Trees T (P −nM)In this section we study the inverse under P of the tree T (M).2.1 Proposition.

P −1T (M) = T (P −1M) = T (P −1VT (M)). In this case the verticesof the tree are given by VT (P −1M) = P −1VT (M).Proof.

As P −1M ⊂P −1VT (M) we have T (P −1M) ⊂T (P −1VT (M)).From Lemma 1.10, PT (P −1VT (M)) = [PP −1VT (M)]K = [VT (M)]K = T (M). It followsthat T (P −1VT (M)) ⊂P −1T (M).Now let z ∈P −1T (M) −P −1M, then P(z) belongs to a regulated arc γ(I) ⊂T (M),with only end points in M. By Lemma 1.14 any inverse of this regulated arc is also regulatedwith endpoints in P −1M and therefore belongs to T (P −1M); in this way z ∈T (P −1M).9

If z ∈P −1M then by definition z ∈T (P −1M). This completes the proof of the chain ofinequalities.The second part follows from the first together with the definition of VT (P −1M) andProposition 1.16.#Proposition 2.1 extends easily.2.2 Corollary.

P −nT (M) = T (P −nM) = T (P −nVT (M)). In this case the vertices ofthe tree are given by VT (P −nM) = P −nVT (M).#2.3 As T (M) ⊂P −1T (M) there are two incidence functions, ν0,M = νT (M) for T (M)and ν−1,M = νT (P −1M) for P −1T (M).

It is immediate that ν0,M(z) ≤ν−1,M(z) at everypoint of T (M). Furthermore, we have the following (here degzP denotes the local degree ofP at z).2.4 Lemma.

ν−1,M(z) = ν0,M(P(z)) degzP, for every z ∈P −1T (M).Proof. This follows from Lemma 1.10 and Proposition 2.1.#These inequalities can be easily generalized for the incidence functions ν−n,M of thetrees P −nT (M).

For example, ν−n,M(z) ≤ν−n−1,M(z) at every point of P −nT (M).The next proposition is a weak attempt to reconstruct the tree P −1T (M) starting fromT (M). An improved version will be given in Chapter III (compare Proposition III.2.5).2.5 Proposition.

Let X be a component of T ∗(P −1M). Denote by C(X) = Ω(P) ∩Xthe critical points in X.

Then P induces a homeomorphism between X and the componentTα of T (M) cut along P(C(X)) that contains P(X).Proof. By Lemma 1.13 P|X is injective.

Also, P(X) is relatively open in Tα. As it isalso compact it must be the whole component.#10

3. Incidence.In this Section we take a closer look at terminal, incidence, branching and non branchingpoints of the Postcritically Finite Polynomial P. A point p ∈J(P) is terminal if there isonly one external ray landing at p. Otherwise p is an incidence point.

For incidence pointswe distinguish between branching (if there are more than two rays landing at p) and nonbranching (exactly two rays landing at p).We will show that for a postcritically finitepolynomial P, every branching point must be periodic or preperiodic. Also we will provethat every periodic branching point is present as a preferred point (see §1.15) in the minimaltree T (M0), and thus in any tree T (M).3.1 Let P be a Postcritically Finite Polynomial, and z an arbitrary point in the Juliaset J(P).

Every component of J(P) −{z} is eventually mapped onto the whole Julia set,and therefore contains points whose orbit contains any specified point. We will use this factin the following two propositions.3.2Proposition.LetPbeaPostcriticallyFinitePolynomialandz ∈J(P) a branching point.

Then z is preperiodic (or periodic).Proof. Suppose z does not eventually map to O(Ω(P)) (otherwise z is already preperi-odic).

Fix w ∈Ω(P) and pick in every component of J(P)−{z} a point pi which eventuallymaps to w. The orbit O({p1, . .

. , pk}) of this set {p1, .

. .

, pk} is a finite set. In this way,the set M ′ = M ∪O({p1, .

. .

, pk}) is invariant and contains the critical points of P. Asz ∈VT (M′), the result follows from Proposition 1.16.#3.3 Proposition. Let P be a Postcritically Finite Polynomial and z ∈J(P) a periodicincidence point.

Then z ∈T (M0), and in this way z ∈T (M) for any finite invariant setM ⊃Ω(P). Furthermore, ν0,M(z) is independent of M and equals the number of componentsof J(P) −{z}.

In particular there are exactly ν0,M0(z) external rays landing at z.Proof. The number of rays landing at z equals the number of components of J(P)−{z}.After this remark the proof is analogous to that of last proposition.

Further details are leftto the reader (compare also Lemma 1.10 and Remark 1.11).#3.4 Corollary. Let z ∈J(P) ∩T (M) be such that P ◦n(z) is periodic.

Then ν−n,M0(z)equals the number of components of J(P) −{z}. In particular there are exactly ν−n,M(z)external rays landing at z.Proof.

This follows from Proposition 3.3 and Lemma 2.4.#3.5 Corollary. T(M) contains a fixed point of P.11

Proof. If P has a fixed critical point, then such point is in M and by definition inT (M).

Otherwise, as there are only d −1 fixed rays, but d fixed points, one must be anincidence point. By Proposition 3.3, this fixed point is in T (M).#12

Chapter IIAbstract Hubbard Trees.In this Chapter we set our basic abstract framework. We carefully justify the impor-tance of all the elements in the definition of abstract Hubbard Trees given in the introduction(compare Examples 2.11-13).

In Section 1, we introduce some basic notation related to finitetopological trees. In Section 2, we introduce dynamics in finite topological trees, and explainwhy further structure should be added in order to have a characterization of postcriticallyfinite polynomials.

In Section 3, the elements needed for this characterization are defined.In Section 4, we give a normalization in order to simplify notation, and we state our mainresult, namely necessary and sufficient conditions for the realization of Hubbard Trees.1. Cyclic Trees.In this Section we only introduce some notation related to finite topological trees whichwould be used throughout the rest of this work.1.1 Definition.

By a topological tree T will be meant a finite connected acyclic m-dimensional simplicial complex (m = 0, 1). Given p ∈T we define the incidence numberνT (p) of T at p as the number of connected components of T −{p}.

We say that p ∈T isa branching point if νT (p) > 2, and an end if νT (p) = 1.A homeomorphism γ : I = [0, 1] →T is called a regulated path in T. In general wewill not distinguish between the map γ and its image γ(I). This because given two pointsp, p′ ∈T , any regulated path joining them will have the same image, which we denote by[p, p′]T .

Given X ⊂T we denote by [X]T the smallest subtree of T which contains X.Clearly this notation is compatible with that introduced before.1.2 Definition. A cyclic tree is a triple (T, V, χ), where(a) T is the underlying topological tree;(b) V ⊂T is finite set of vertices so that each component of T −V is an open 1-cell(an edge);(c) For each v ∈V , χv represents a cyclic order in the set Ev = {ℓ1, .

. .

, ℓk} of all edgeswith v as a common endpoint.13

The presence of these χv naturally determines an isotopy class of embeddings of thistree T into C.1.3 Pseudoaccesses. If ℓ, ℓ′ ∈Ev are consecutive in the cyclic order of Ev, we saythat (v, ℓ, ℓ′) is a pseudoaccess to v. Take a pseudoaccess (v, ℓ, ℓ′) to v, and let the end pointsof the edge ℓ′ be v, v′ ∈V .

At Ev′ let ℓ′′ be the successor of ℓ′ in the cyclic order. We saythat (v′, ℓ′, ℓ′′) is the successor of (v, ℓ, ℓ′).1.4 Lemma.

Let (T, V, χ) be a cyclic tree. The successor function in the set of pseu-doaccess to the vertices in V is a complete cyclic order.Proof.

A trivial induction in the cardinality of V .#1.5 Remark. A Postcritically Finite Polynomial P and a finite invariant set M con-taining the critical set Ω(P) of P, naturally defines a cyclic tree (T (M), VT (M), χ).

Here χvrepresents the cyclic order of the components around a point v ∈VT (M) taken counterclock-wise.1.6 Definition. Let (T, V, χ) be a cyclic tree, and let M ⊂V .

We define the restrictionof (T, V, χ) to M, as the cyclic tree ([M]T , VM, χ′) where VM is the union of M and thebranching points in the topological tree [M]T , and χ′v is the natural restriction of the cyclicorder χv of Ev to the set E′v of all edges of [M]T incident at v.2. Dynamical Abstract Trees.In this Section we give our first attempt to describe the dynamics of a PostcriticallyFinite polynomial by means of the dynamics in a finite topological tree.Unfortunatelythis simple characterization proves to be weak (compare Examples 2.11-13), and furtherstructure has to be added.

This will be done in Sections 3 and 4.2.1 Definition. A dynamical abstract tree is a triple T = ((T, V, χ), τ, δ) where(a) (T, V, χ) is the underlying cyclic tree,(b) τ : V →V is the vertex dynamics,(c) δ : V →Z is a positive local degree function.We require these elements to be related as follows,(i) For any edge ℓwith endpoints v, v′ ∈V we must have τ(v) ̸= τ(v′).14

This condition allows us to extend τ to the underlying tree as follows. For any edgeℓwith endpoints v, v′ ∈V , map ℓhomeomorphically to the shortest path joining τ(v) andτ(v′).

Any extension ‘τ’ well defines a map τv : Ev →Eτ(v). We require,(ii) For any v ∈V , there exists a cyclic ordered set Ev such that Ev embeds in anorder preserving way into Ev.

We require that τv can be extended to a degree δ(v) orien-tation preserving covering map between Ev and Eτ(v) (see appendix B). For the practicalinterpretation of this set Ev we refer to Remark 2.2 and Proposition III.2.5.We define the degree of T as deg(T) = 1 + Pv∈V (δ(v) −1).

We require(iii) deg(T ) > 1.2.2 Remark. A Postcritically Finite Polynomial P of degree n > 1 and a finite invari-ant set M ⊃Ω(P) naturally defines a dynamical abstract tree TP,M = ((T (M), VT (M), χ), P, degzP)of degree n. Here Ev represents the components around v in the tree T −1M (see §I.2).2.3 Definitions.

Let T = ((T, V, χ), τ, δ) be a dynamical abstract tree. We extend δto all the tree T by letting δ(p) = 1 if p ̸∈V .

We define the critical set of T as Ω(T) ={p ∈T : δ(p) > 1}. Condition (iii) above implies that Ω(T) is always non empty.

A pointp ∈Ω(T) is a critical point; otherwise, it is non critical.The orbit of S ⊂V is the set O(S) = ∪∞k=0τ ◦k(S).2.4 Definition. Let ℓ∈Ev, we denote by Bv,T (ℓ) the closure of the connected com-ponent of T −{v} that contains ℓ.

This is just the branch at v determined by ℓin the treeT .2.5 Definition. Let T = ((T, V, χ), τ, δ) be an abstract tree, and let M ⊂V be aninvariant set of vertices containing the critical set (τ(M) ∪Ω(T) ⊂M).

We define the re-striction T(M) of T determined by M, as the abstract tree T(M) = ((T (M), VM, χ), τ′, δ′),where (T (M), VM, χ) is the restriction of the angled tree as defined in §1.8, and τ ′, δ′ arerestrictions of the functions τ, δ to the set VM.2.6 Definition. Let T, T′ be two abstract trees of degree n = deg(T) = deg(T′) > 1.We say that T′ is an extension of T (in symbols T ⪯T′), if there is an embedding φ : T →T ′which satisfies the obvious conditions:(i) φ(V ) ⊂V ′,(ii) τ ′(φ(v)) = φ(τ(v)) and(iii) δ(v) = δ′(φ(v)) for all v ∈V ,(iv) φ induces a cyclic order preserving embedding of Ev into Eφ(v).

(At this point itis convenient to think of the elements of Ev as ‘germs of edges’. )15

Clearly ⪯is an order relation.2.7 Let T, T′ be two abstract trees of degree n = deg(T) = deg(T′) > 1. We say thatT′ is equivalent to T (in symbols T ≈T′), if T ⪯T′ and T′ ⪯T.

This determines anequivalence relation between abstract trees. Furthermore, the order relation ⪯extends to apartial order between equivalence classes of dynamical abstract trees of degree n > 1.2.8 Definition.

We say that an equivalence class [T] of dynamical abstract trees ofdegree n > 1 is minimal if given [T′] ⪯[T] we necessarily have [T′] = [T].From the definition of extension tree we can deduce that if [T′] is an extension of [T],then [T] is a restriction of [T′] in the sense of Definition 2.5. Therefore we have the following.2.9 Proposition.Every abstract tree T contains a unique minimal tree min(T).Furthermore, this unique minimal tree is the tree generated by the orbit O(Ω(T)) of thecritical set.#2.10 The question now is if this description completely characterizes PostcriticallyFinite Polynomials.

In other words, given a class [T] of dynamical abstract trees, is therea unique (up to affine conjugation) Postcritically Finite Polynomial P and an invariant setM ⊃Ω(P) such that TP,M ∈[T]?The answer is negative as the following examples show.2.11 Non uniqueness. Suppose a degree 3 polynomial has the following minimal treeT (where the double star stands for a double critical point, i.e, its local degree is 3).•x1⋆⋆x0•x2=x3Figure 2.1.

The vertex dynamics is given by x0 7→x1 7→x2 7→x3 = x2.If we want a centered monic polynomial with this minimal tree we suppose that x0 = 0.We have then P(z) = z3 + c. (For polynomials of the form Pc(z) = z3 + c, the number c2is a complete invariant up to conjugacy. In other words Pc is affine conjugate to Pc′ if andonly if c2 = c′2.) If P has this minimal tree, then the orbit of the critical point is as follows,0 7→c 7→c3 + c 7→c3 + c.In this way, the relation P ◦2c (0) = P ◦3c (0) determines the equation c3 +c = (c3 +c)3 +c.Thus c must satisfy c5(c4 + 3c2 + 3) = 0.

If we want c3 + c ̸= 0 we must have c ̸= 0, andwe have two different possible values for c2 = −3±√−32. For both values of c2 the respective16

polynomials Pc have minimal tree T (M0) as shown in Figure 2.1. In fact, by Lemma I.1.13,c and c3 + c belong to different components of T −{0}.In this way, we have constructed two different non affine conjugate polynomials P, P ′which define the same class of minimal trees.

Nevertheless, the trees TP,OP −1Ω(P ), TP ′,OP ′−1Ω(P ′)belong to different classes (see Figure 2.2 below).• x−12x1•x−10•x0⋆⋆x−10•x2=x3•• x−10• x−10•x1•x−10⋆⋆x0•x−10•x2=x3• x−12Figure 2.2. Here x−1jmaps to xj.

Even if the trees are isomorphic, they fail to havethe same cyclic order around x0.2.12 Non existence (compare Figure 2.3). The class of the tree below can not beobtained from a polynomial map.

It must correspond to a degree two polynomial with threefixed points, which is impossible.•y0=y1∗x0=x1•z0=z1Figure 2.3. All vertices are fixed.

Here δ(x0) = 2, and δ(y0) = δ(z0) = 1.Here is an alternative description of the obstruction for ‘realizing’ this tree. If thistree is equivalent to a tree TP,M of a degree two polynomial P, edges whose commonvertex is the Fatou critical point must be realized near this vertex as internal rays in theuniformizing coordinate (see Section I.1.1).

Let α be the difference of the two arguments inthis coordinate. We have then 2α = α + 1 (mod 1).

But this implies that the two segmentsshould be identified. Note that the minimal tree corresponding to this tree has only x0 asvertex.

Thus, this minimal tree can be realized as Tz7→z2,{0}.2.13 Non existence (compare Figure 2.4). The class of the minimal tree below cannot be obtained from a polynomial map.

If there is a tree TP,M in the class of such tree, itwill not satisfy the expanding property (compare Propositions I.1.18 and I.1.21).17

z1y1y =y0 2z =z0 2x =x0 2x1**Figure 2.4. For any k ≥0 there is no vertex between τ◦k(x0) and τ ◦k(x1).2.14 All that can go wrong already happened in these three examples.

Uniquenessfailed because we had too little information. Here too little information means that we donot have enough information to recover in a unique way the tree T −1(M) (see Section I.2,compare also Propositions I.2.5 and III.2.5).

Examples 2.12 and 2.13 failed because they donot satisfied necessary conditions. Namely, the trees must have well defined angles aroundFatou critical points (see Section I.1) and should satisfy the expanding condition betweenJulia type vertices (see Proposition I.1.21).3.

Angled Trees.In this section we introduce the class of trees that will model our results. In this classwe must be able to consider an analogue of the expanding condition, and also to defineangles between edges near Fatou points.3.1 Definition.

An angled tree is a pair A = (T, ̸ ), where(a) T = ((T, V, χ), τ, δ) is a dynamical abstract tree,(b) together with a function ℓ, ℓ′ 7↛ (ℓ, ℓ′) = ̸v(ℓ, ℓ′) ∈Q/Z which assigns a rationalmodulo 1 to each pair of edges ℓ, ℓ′ which meet at a common vertex. This angle ̸ (ℓ, ℓ′)should be skew-symmetric, with ̸v(ℓ, ℓ′) = 0 if and only if ℓ= ℓ′, and with ̸v(ℓ, ℓ′′) ≠v(ℓ, ℓ′) + ̸v(ℓ′, ℓ′′) whenever three edges are incident at a vertex v.The maps ̸ , τ and δ must be related as follows.

Again we extend τ to a map τ : T →Twhich carries each edge homeomorphically onto the shortest path joining the images of itsendpoints. Any extension well defines a map between ‘germs’ τv : Ev →Eτ(v).

We requirethen that̸τ(v)τv(ℓ), τv(ℓ′)= δ(v)̸vℓ, ℓ′,(1)whenever ℓ, ℓ′ ∈Ev (in this case τv(ℓ), τv(ℓ′) contain edges incident at τ(v) where the anglebetween them is measured).18

Such an angle function determines a cyclic order in Ev which we suppose to coincidewith χ. Note that in this case the angle function ̸v at v can be extended to a bigger set Ev(see §3.3 below).The degree deg(A) of the angled tree A = (T, ̸ ) is by definition the degree of theabstract tree T. The critical set Ω(A) of A is by definition Ω(T).3.2 A vertex v ∈V is called periodic if for some m > 0 we have τ◦m(v) = v. The orbitof a periodic critical vertex is a critical cycle.

We say that a vertex v is of Fatou type (or aFatou vertex) if eventually maps to a critical cycle. Otherwise it is of Julia type (or a Juliavertex).

If v0 7→v1 7→. .

. 7→vm = v0 is a critical cycle, we define the degree of the cycle asthe product δ(v0) × .

. .

× δ(vn−1) of the degrees of the elements in said cycle.3.3 The function τ induces a function τv between the set Ev of edges incident at vand the set Eτ(v) of edges incident at τ(v).Given a Fatou periodic vertex we can findembeddings φv →R/Z called local coordinates of the set Ev (see Appendix B) such thatthe diagramEvτv−→Eτ(v)φvyyφτ(v)R/Zmv−→R/Z(2)commutes. Here mv is multiplication by δ(v) (modulo 1).

Note that the number of possibleembeddings for each critical cycle is the degree of the cycle minus one.At other Fatou vertices v we can still make diagram (2) hold by pulling back the localcoordinate at τ(v) and using relation (1).At periodic Julia vertices relation (1) easily implies that τv is a bijection. We pick anelement ℓ∈Ev to which we assign the 0 coordinate (φv(ℓ) = 0).

If Eτ(v) has not beenassigned a local coordinate, we assign to each edge τv(ℓ) ∈Eτ(v) the argument φv(ℓ). Ingeneral we can not make diagram (2) commute for all vertices.

(It might fail at the startingvertex v). In this last case the induced function mv in R/Z becomes translation by someconstant.At non periodic Julia vertices a local coordinate φv can be pulled back from Eτ(v) inδ(v) different ways such that diagram (2) commutes.3.4 Definition.

Let A = (T = ((T, V, χ), τ, δ), ̸ ) be an angled tree. For a finite set ofinvariant vertices M ⊃Ω(A), we denote by A(M) = (T(M), ̸M) the angled tree generatedby M, i.e, take T(M) the dynamical abstract tree determined by M (see section 2.5), andlet ̸M be the restriction of ̸to the vertices of T(M).

Of course, A = A(V ).3.5 Lemma. For any extension ‘τ’ and invariant set of vertices M ⊃Ω(A) we haveτ(T (M)) = [τ(M)]T .19

Proof.A copy of Lemma I.1.10 with the appropriate change of notation (see alsoLemma 3.7).#3.6 Definition. Let A be an angled tree of degree n, and let Ω(A) = {v1, .

. .

, vl} bethe critical set. For a fixed family of local coordinates {φ}v∈V , we construct a partitionT ∗= T ∗({φ}) of T consisting (counting possible repetitions) of exactly n subtrees of T .This partition will have the property that every point p ∈T will belong to exactly δ(p)elements of T ∗.

Note that this will be possible only if we somehow ‘unglue’ the tree aroundevery critical point. This is formally done as follows (compare the example within the proofof Proposition III.2.5).Let T0 be {T }.

We will inductively define partitions Ti (i ≤l) of T with the followingproperties(a) For j ≤i, vj belongs to exactly δ(vj) elements of Ti,(b) For j > i, vj belongs to exactly one element of the family Ti,(c) Ti is constructed from Ti−1 by replacing the unique element T (α) of Ti−1 to whichvi belongs by δ(vi) subtrees of T (α).We proceed as follows. Let T (α) be the only element of Ti−1 to which vi belongs.

Wepartition T (α) into δ(vi) pieces as follows. First divide the set Ei = Evi in δ(vi) subsetsusing the local coordinate.

For this we define for k = 0, . .

. , δ(vi) −1,Ek = Eki = {ℓ∈Ei : φvi(ℓ) ∈[kδ(vi), k + 1δ(vi) )}Now, we take the union of all branches in a set Ek, i.e, defineT k(α) = T (α) ∩vi ∪[ℓ∈EkBvi,T (ℓ).Define now Ti by removing T (α) of the family and including all such T k(α).

By definitionT ∗is the last partition Tl.3.7 Lemma. Let A be an angled tree.

Then the vertex dynamics τ induces a continuousmap of T into itself, where the restriction to every element (component) of T ∗is injective.Proof. (Compare Lemma I.1.8.

)Let Tα be an element of T ∗.Suppose there aredifferent p1, p2 ∈Tα so that τ(p1) = τ(p2). Take a path γ : I →[p1, p2]T ⊂Tα joiningp1, p2.

As τ|Tα is locally one to one, the set ∆= {(t1, t2) : t1 < t2 and τ(γ(t1)) = τ(γ(t2))}is compact. As we have assumed that this set is not empty we can take (t1, t2) ∈∆witht2 −t1 minimal.

Let t ∈(t1, t2), then τ(γ([t1, t])) and τ(γ([t, t2])), are regulated arcs withthe same end points. Therefore they are equal and thus t2 −t1 is not minimal.#3.8 Remark.

As T ∗consists of n = deg(A) elements (counting possible repetitions),it follows from the last lemma that for any p ∈τ(T ) and any possible extension ‘τ’X{q∈T :τ(q)=p}δ(q) ≤n.20

3.9 Lemma. Let v be a periodic Fatou vertex, and ℓ1, ℓ2 ∈Ev be different edges.

Thereis an n ≥0 so that τ ◦nv (ℓ1) and τ ◦nv (ℓ2) belong to different components of T ∗.Proof. Let d > 1 be the degree of the cycle v0 = v 7→v1 7→.

. .

7→vm = v0. We writeφv(ℓ1) and φv(ℓ2) in base d expansion.

If for all n, τ◦nv (ℓ1) and τ ◦nv (ℓ2) belong to the samecomponent of T ∗, by construction for all k > 0 the integer parts of mδ(vk)φvk(τ ◦k(ℓ1)) andmδ(vk)φvk(τ ◦k(ℓ2)) are equal. But this implies that ℓ1 = ℓ2.#3.10 Definition.

(Compare §I.1.20.) We define the distance dT (v, v′) between verticesas follows.

Set dT (v, v) = 0. Otherwise let dT (v, v′) be the number of edges between v andv′.We say that the angled tree A = (T, ̸ ) is expanding if the following property is satisfied(see also Propositions I.1.18 and I.1.21).For any edge ℓwhose end points v, v′ are Julia vertices there is an m ≥1 such thatdT (τ ◦m(v), τ◦m(v′) > 1.Equivalently, A is not expanding if and only if there exists periodic Julia vertices v, v′such that dT (τ ◦m(v), τ◦m(v′)) = 1 for all m ≥0.3.11 Lemma.

An angled tree A is expanding if and only if for any two periodic Juliavertices v, v′ there is an m ≥0 such that τ◦m(v) and τ◦m(v′) belong to different componentsof T ∗.Proof. Suppose A is not expanding.

By definition there are periodic Julia verticesv, v′ with dT (τ ◦m(v), τ◦m(v′)) = 1 for all m ≥0. As there are no critical points in the orbitof periodic Julia vertices, by construction τ ◦m(v) and τ ◦m(v′) will be in the same elementof T ∗for any possible choice of the family {φv}.Let now A be expanding.

Suppose there are different Julia vertices v, v′ such thatτ ◦m(v),τ ◦m(v′) belong to the same component of T ∗for all m ≥0. Among such pairs wecan take v, v′ periodic and with the property that dT (v, v′) is minimal.

By assumption theregulated path [τ ◦m(v)τ ◦m(v′)]T is completely contained within a component of T ∗for allm ≥0. It follows from Lemma 3.7 that all [τ◦m(v)τ ◦m(v′)]T are homeomorphic.

We takev′′ ∈[v, v′]T ∩V such that dT (v, v′′) = 1. As A is expanding it follows that v′′ is a periodicFatou vertex.

In this way Ev′′ ∩[v, v′]T = {ℓ1, ℓ2} with ℓ1 ̸= ℓ2. We get a contradiction inapplying Lemma 3.9.#3.12 Corollary.

Let A be an expanding angled tree. The induced angled tree A(M) isexpanding for every invariant set of vertices M ⊃Ω(A).#3.13 Lemma.

Let A be an expanding angled tree. Given a periodic Julia vertex v,every component of T −{v} contains a vertex which belongs to O(Ω(A)).21

Proof.Suppose that Bv,T (ℓ) does not contain a vertex in O(Ω(A)) different fromv for some ℓ∈Ev.The relation τv(ℓ) ∈Eτ(v) determines a cyclic sequence of edgesℓ= ℓ0 ∈Ev, ℓ1 ∈Eτ(v), . .

. , ℓn = ℓ0 ∈Eτ ◦m(v) = Ev.If for some k < m the branchBτ ◦k(v),T (ℓk) contains a critical point, we may assume that k is as big as possible andderive a contradiction by using Lemma 3.7.

We assume though that Bτ ◦k(v),T (ℓk) does notcontains a critical point for all k. This implies using again Lemma 3.7 that all Bτ ◦k(v),T (ℓk)are homeomorphic with only periodic Julia vertices. Thus, A is not expanding.#4.

Abstract Hubbard Trees.The angles at Julia vertices are rather artificial, so we normalize them as follows. If medges ℓ1, .

. .

, ℓm, meet at a periodic Julia vertex v, then we assume that the angles ̸v(ℓl, ℓk)are all multiples of 1/m (it follows that the angles at periodic Julia vertices convey noinformation beyond the cyclic order of these m incident edges). Fortunately, this number ispreserved under restrictions which contain the orbit of the critical set.

This will allow us togive a coherent description.4.1 Definition.By an abstract Hubbard Tree we mean an expanding angled treeH = (T, ̸ ) such that the angles at any periodic Julia vertex where m edges meet aremultiples of 1/m.4.2 Let H, H′ be two abstract Hubbard Trees of degree n = deg(H) = deg(H′) > 1. Wesay that H′ is an extension of H (in symbols H ⪯H′), if there is an embedding φ : T →T ′which satisfies the obvious conditions:(i) φ(V ) ⊂V ′,(ii) τ ′(φ(v)) = φ(τ(v)) and(iii) δ(v) = δ′(φ(v)) for all v ∈V ,(iv) ̸v(ℓ, ℓ′) = ̸′φ(v)(φ(ℓ), φ(ℓ′)) for all ℓ, ℓ′ ∈Ev.Clearly ⪯is an order relation.4.3 Let H, H′ be two abstract Hubbard Trees of degree n = deg(H) = deg(H′) > 1.We say that H′ is equivalent to H (in symbols H ∼= H′), if H ⪯H′ and H′ ⪯H.ThisdeterminesanequivalencerelationbetweenabstractHubbardTrees.

Furthermore, the order relation ⪯well defines a partial order between equivalenceclasses of abstract Hubbard Trees of degree n > 1.4.4 Lemma. Let H be an abstract Hubbard Tree, and M ⊃Ω(H) a finite invariant setof vertices.

Then H(M) is an abstract Hubbard Tree and H(M) ⪯H.22

Proof. This follows from Corollary 3.12 and Lemma 3.13.#4.5 Proposition.Every abstract Hubbard Tree H contains a unique minimal treemin([H]).

Furthermore, this unique minimal tree is the tree generated by the orbit O(Ω(H))of the critical set.Proof. This follows from Proposition 2.8 and Lemma 4.4.#4.6 Remark.

A Postcritically Finite Polynomial P and a finite invariant set M ⊃Ω(P)naturally defines an abstract Hubbard Tree HP,M = (TP,M, ̸ ). To define the angle functionwe note the following.

At Fatou periodic vertices the edges of the tree are by definitionsegments of constant argument in the B¨ottcher coordinate (see I.1.1), we define the anglebetween two such edges as the difference of their coordinates. For other Fatou points thecoordinate can be defined such that the diagram (2) commutes, and we proceed as above.For a Julia set point v, J(P) −{v} consists of a finite number (say m) of components.

Wedefine the ‘angle’ between these components to be a multiple of 1/m. As edges in the treecorrespond locally to some of these components we have an angle function between them.

(This procedure is well defined and compatible with the definition above, see PropositionI.3.3). It is easy to see that the minimal tree TM0 (see I.1.9) corresponds to the minimaltree HP,M0 = min(HP,M) of any bigger invariant set M.The main result of this work is the following.4.7 Theorem.

Let H be an abstract Hubbard Tree. Then there is a unique (up toaffine conjugation) Postcritically Finite polynomial P, and an invariant set M ⊃Ω(P) suchthat HP,M ∈[H].4.8 Theorem.

Equivalence classes of minimal abstract Hubbard Trees of degree n > 1are in one to one correspondence with affine conjugate Post-critically finite polynomials.We prove Theorem 4.7 in the next chapter. Theorem 4.8 is an easy consequence of thisresult and Proposition 4.5.23

Chapter IIIRealizing Abstract Hubbard Trees.In this chapter we give the proof of the realization Theorem for Abstract HubbardTrees (Theorem II.4.7). Our proof depends in the theory of Critical Portraits developedin the first part of this work.In Section 1 we define the class of extensions which donot add any essential information to the tree.

We will prove later that every extensionbelongs to this class (compare Corollary 4.6). Section 2 gives the abstract analogue of §I.2,where we show that a Hubbard Tree contains all the information required to reconstruct its‘inverse’.

Section 3 gives the abstract analogue of §I.3. In Section 4 we relate the ‘accessesto Julia points’ with the argument of a possible ‘external ray’ (compare Theorem B in theintroduction).

As a consequence of this, we prove that every extension of a Hubbard Treeis canonical in the sense described in Section 1. In Section 5 we associate a Formal CriticalPortrait to our Tree.

This Critical Portrait is also admissible as shown in Section 6. Finallywe prove that the Hubbard Tree associated with this critical portrait is equivalent to thestarting one, thus establishing the result.

From now on, we omit the trivial case in which Tis a single critical vertex.1. Canonical Extensions.In this Section we define what we call ‘canonical extensions’.

We will prove in Section4 that every extension which itself is a Hubbard Tree, is canonical in the sense describedhere. This fact will allow us later to associate in a natural way a critical portrait to everyHubbard Tree.1.1 Definition.Let H0 ⪯H1 be abstract Hubbard Trees.We say that H1 is acanonical extension of H0 if for every extension H ⪰H0, there is a common extension of Hand H1.

Canonical extensions always exist. By definition every Hubbard Tree is a canonicalextension of itself.

Our final goal in this direction will be to prove that every extension iscanonical (compare Corollary 4.6).1.2 Proposition. Let H be an abstract Hubbard Tree and ω a periodic Fatou vertex.There is a canonical extension H′ of H such that(a) Ev = E′v at all vertices of the original Hubbard Tree H.24

(b) For every periodic ℓ∈E′ω with end points ω, v in H′, the vertex v is of Julia typeand dH′(τ ◦m(ω), τ◦m(v)) = 1 for all m ≥0.In fact, the underlying topological trees can be chosen to be the same, with only newJulia vertices to be added.Proof. Suppose the edge ℓhas end points ω, v, and its germ is of period k. In otherwords suppose the induced maps τv determine a periodic sequence of edges ℓ0 = ℓ∈Eω, ℓ1 ∈Eτ(ω), .

. .

, ℓk = ℓ0 ∈Eτ ◦k(ω) = Eω. We distinguish two cases.Suppose dH(τ ◦m(ω), τ◦m(v)) = 1 for all m.If v is of Julia type, condition (b) isalready satisfied.

If v is of Fatou type then by Lemma II.3.7 all ℓk = [τ◦m(ω), τ◦m(v)]T arehomeomorphic. In this case we insert a vertex vm in each ℓm (if ℓm = ℓl then vm = vl)and define τ(vm) = vm+1.

Then clearly v1 is periodic of period k or k/2. The angles at vkare 1/2 because two edges will meet now.

Note that in this case this is the only possibleextension that involves the segments [τ ◦m(ω), τ◦m(v)]T and gives an expanding tree.Otherwise, suppose dH(τ ◦m(ω), τ◦m(v)) > 1 for some m ≥1. In this case we insert avertex vm in each ℓm as close as possible to τ ◦m(ω) (note here that if ℓm = ℓl then we musthave vm ̸= vl) and define τ(vm) = vm+1.

Clearly v1 is periodic of period k. The angles atvk are 1/2 because two edges meet now.The only obstruction to this construction is if condition (b) is already satisfied. There-fore the extension is canonical.#1.3 Corollary.

Every abstract Hubbard Tree has a canonical extension with at leastone Julia vertex.#2. Inverse Hubbard Trees.We now describe an important type of canonical extension.

In the case of the HubbardTree HP,M generated by a polynomial P and an invariant set M, the interpretation issimple. We will reconstruct the equivalence class of the abstract Hubbard Tree generatedby P −1M starting from HP,M.

Thus, this section is the abstract analogue of Section I.2.2.1Definition.AnabstractHubbardTreeHofdegreen>1ishomogeneous if(a) ∀v ∈τ(V ), n = P{v′∈V :v=τ(v′)} δ(v′), and(b) Ω(H) ⊂τ(V ).In other words, every vertex with at least one inverse must have a maximal numbercounting multiplicity (compare Remark II.3.8). Furthermore, all critical vertices must have25

a preimage. The terminology is justified by the fact (proved below) that the underlyingtopological tree can be ‘chopped’ into n pieces; each piece being homeomorphic as a graphto the abstract tree generated by restriction to τ(V ).

More formally, τ establishes a home-omorphism between each of the n elements of T ∗(compare Section II.3.6) and the abstractHubbard Tree H(τ(V )).2.2 Lemma. For any election of local coordinate system {φv}v∈V , each Tα ∈T ∗({φv})is homeomorphic to H(τ(V )).Proof.

By Lemma II.3.5 we have τ(T ) = [τ(V )]T . Also, every v ∈[τ(V )]T has at mostone inverse in Tα by Lemma II.3.7.

It follows easily from condition a) that every v ∈[τ(V )]Tshould have a unique inverse in Tα. The result follows.#2.3 Corollary.

Let H be an abstract tree of degree n > 1, such that Ω(H) ⊂τ(V ). ThenHishomogeneousifandonlyif#(V )−1=n(#(τ(V )) −1).Proof.

This follows from Lemma 2.2 and Remark II.3.8.#2.4 Definition. Let H′ ⪯H be abstract Hubbard Trees with H homogeneous.

Wesay that H′ is the image of H if the embedding which defines the order H′ ⪯H is such thatalso H(τ(V )) ∼= H′.This definition clearly extends to equivalence classes of abstract Hubbard Trees.2.5 Proposition. Every equivalence class [H] of abstract Hubbard Trees is the imageof a unique class of homogeneous abstract Hubbard Trees.Proof.

The proof of existence is constructive using only necessary conditions, unique-ness follows. Let {φv}v∈V be a family of local coordinates for V ⊂T .

We will work with thefamily T ∗= T ∗({φv}) (compare §II.3.6). We construct a new simplicial complex by gluing adifferent copy of T to each component Tα ∈T ∗following τ (compare Lemma 2.2).

In otherwords we consider n disjoint copies Hα of T (α = 1 . .

. n), with a suitable identificationat “critical points” described below.

By Lemma II.3.7, the dynamics τ restricted to eachsubtree Tα of the family T ∗= T ∗({φv}) is one to one. We denote this restriction by iα(“i” stands for identification).

Thus we have a family of maps iα : Tα →Hα. We establishan equivalence relation ∼between points in the disjoint union ` Hα as follows.

Wheneverω ∈Tα ∩Tβ (and this can only happen if ω is critical), we write iα(ω) ∼iβ(ω). Thus thenew underlying topological tree is X = ` Hα/ ∼.

There is a ‘natural inclusion T ⊂X’induced by the maps iα. The new set of vertices is the disjoint union of vertices of Hαmodulo ∼.26

In order to avoid confusion in the above notation, we will interrupt the proof in orderto exemplify our construction.•x1∗∗x0∗x2 = x3The abstract tree in the figure above can be chopped into 4 pieces according to the con-struction in §II.3.6. We think of these pieces as mapping onto different copies Hα of T (thisis emphasized below by the superscripts in the right).T1 :•x1•x0i1−→•x11•x10•x12T2 :•x0i2−→•x21•x20•x22T3 :•x0•x2=x3i3−→•x31•x30•x32T4 :•x2i4−→•x41•x40•x42In this way the new tree is given by identifying x11 = i1(x0), x21 = i2(x0) and x31 = i3(x0)(because x0 ∈T1 ∩T2 ∩T3) and by identifying x32 = i3(x2) with x42 = i4(x2) (becausex2 ∈T3 ∩T4).

Note that the original tree is canonically embedded in this new one by usingiα.Proof of 2.5 (Continue). We continue the proof by defining the dynamics and anglefunctions.

What we have done so far is simply to replace each piece Tα by the copy Hα.In this way, if we think of the Hα as the corresponding pieces for the new tree, the finalstructure is induced by the old one by gluing the Hα following that same pattern of the Tα.The vertex dynamics ¯τ maps each new vertex to the actual point in V from which itwas constructed. More formally, take v ∈Hα a vertex of X; as Hα is also partitioned by thefamily T∗, it follows that v ∈Tβ for some β.

We define ¯τ(v) = iβ(v) ∈Hβ ⊂X. (Clearlythis is well defined and two consecutive vertices have different image).

The degree is one ateach vertex not present in the original tree. In other words, if v ∈Tα (that is if v belongsto the original tree T ), we define the degree at iα(v) (which is the point in X to which v isidentified) as ¯δ(iα(v)) = δ(v).

If v ∈X is not of the form iβ(ω) for some ω, we set ¯δ(v) = 1.The angle function at non critical points is pulled back from the identification: if¯δ(w) = 1, we have a natural homeomorphism between a neighborhood of w ∈X and aneighborhood of w ∈T . The angle function is then copied from the original Hubbard TreeH.

At critical points, it is enough to extend the coordinate functions φv in a compatibleway; the angle between edges can be read from this. We proceed as follows.

Let v ∈Tα becritical. We will define the coordinate φiα(v) at iα(v) ∈X as follows.

By definition (compare27

§II.3.6) there is a k such that ℓ∈Ev belongs to Tα if and only if φv(ℓ) ∈[kδ(v), k+1δ(v)). Now,an edge ℓincident at iα(v) must belong to a unique Hα and therefore corresponds to aunique edge ℓ′ ∈Eτ(v) in the original tree T .

Define φiα(v)(ℓ) =k+φτ(v)(ℓ′)δ(v).As no new periodic vertices are added the tree is still expanding. At periodic Juliavertices no new edges are added (compare §II.3.3).

Therefore, we have a Hubbard Treewhich is homogeneous by Corollary 2.3 and satisfies the required properties.To prove uniqueness, we note that any other local coordinate system {φv}v∈V is alsocanonically present in the new tree constructed. It follows from Lemma 2.2 that the cor-responding partition with respect to this coordinate is independent of the starting localcoordinate system.#2.6 Definition.

Let [H], [H′] be equivalence classes of abstract Hubbard Trees. Wesay that the equivalence class [H′] of homogeneous abstract Hubbard Trees is the inverse of[H] (in symbols inv(H) = H′), if [H] is the image of [H′].Thus, by Propositions 2.3 and 2.5, inv determines a one to one mapping from equiva-lence classes of abstract Hubbard Trees of degree n > 1 to itself.

Furthermore, in this newlanguage Proposition 2.5 reads as follows.2.7 Proposition.Let H be an abstract Hubbard Tree, then inv(H) is a canonicalextension of H.#2.8 Corollary. Let H be an abstract Hubbard Tree and ω a Fatou vertex.

There is acanonical extension H′ of H such that(a) Ev = E′v at all vertices of H.(b) For every ℓ∈E′ω with end points ω, v, we have that v is of Julia type, anddH′(τ ◦k(ω), τ◦k(v)) = 1 for all k ≥0.Proof.We apply first Proposition 1.2 and then take a finite number of ‘inverses’(Proposition 2.5). Finally we restrict to the tree generated by the original vertices.#2.9 Corollary.

Let H be an abstract Hubbard Tree. Then H has a canonical extensionin which all ends are of Julia type.Proof.We apply first Proposition 1.2 and then take a finite number of ‘inverses’(Proposition 2.5).

Finally we restrict to the tree generated by the required vertices.#3. Incidence.28

In this section we study from the dynamical point of view, how the number of edgesincident at a Julia vertex can grow as we take inverses. This section is the abstract analogueof Section I.3.3.1 Definition.

Let [H] be an equivalence class of abstract Hubbard Trees. We definethe incidence number νH(v) at a vertex v ∈V as the number of connected componentsof T −{v} in any underlying topological tree T .

In the inverse trees inv◦m([H]) we havealso incidence functions νH,−m = νinv◦m([H]) at the vertices of inv◦m(H). By definitionνH,0(v) ≤νH,−1(v) for v ∈V .

Also by construction of inv(H), it follows that νH,−1(v) =δ(v)νH,0(τ(v)) for all vertices in inv(H).3.2 Proposition. Let [H] be an equivalence class of abstract Hubbard Trees.

For everyperiodic Julia vertex v ∈V and m ≥0 we have νH,0(v) = νH,−m(v).Proof. As δ(v′) = 1, for every point v′ ∈O(v), no new edges are added around v inthe construction of inv◦m([H]).

(See also Lemma II.3.13. )#3.3 Corollary.

Let [H] be an equivalence class of abstract Hubbard Trees. Let v ∈Vbe a Julia vertex such that τ ◦k(v) is periodic.

Then for every m ≥k we have νH,−k(v) =νH,−m(v).#3.4 Corollary. Let [H] be an equivalence class of abstract Hubbard Trees.

There is ak ≥0 such that for all m ≥k we have νH,−k(v) = νH,−m(v) at every Julia vertex v ∈VH.#We denote such numbers by νH,−∞(v).4. Accesses and External Coordinates.Inthissectionweassociatetoevery‘access’ataJuliavertexanargument.

This coordinate system will allow us to define extensions with ‘reasonable’ prop-erties. Combining these two results we prove that every extension of a Hubbard Tree iscanonical.4.1 Definition.

(Compare Definition II.1.3.) Let H be an abstract Hubbard Tree.Given ℓ, ℓ′ ∈Ev consecutive in the cyclic order, we say that (v, ℓ, ℓ′) is an access to v ifνH,0(v) = νH,−∞(v).

If νH,0(v) < νH,−∞(v) we say that (v, ℓ, ℓ′) is a strict pseudoaccess tov in H. Note that at Fatou vertices there are no possible accesses. Clearly an access at v isperiodic if and only if v is periodic.

These concepts extend to equivalence classes.29

4.2 Lemma. Let H be an abstract Hubbard Tree of degree n. Then τ induces a degreen orientation preserving covering mapping between the pseudoaccesses of the trees inv(H)and H. Furthermore, accesses in inv(H) map to accesses in H.Proof.If(v, ℓ, ℓ′)isapseudoaccessininv(H),byconstruction(τ(v), τv(ℓ), τv(ℓ′)) is a pseudoaccess in H.Clearly this is n to 1, and order preservingby construction.

The second part is obvious.#4.3 Proposition. Let H be a homogeneous abstract Hubbard Tree of degree n > 1 withat least one Julia vertex.

There exist an embedding φH of the accesses of H into R/Z suchthat the induced map between accesses becomes multiplication by n (modulo 1). FurthermoreφH is uniquely defined up to a global addition of a multiple of 1/(n −1).Proof.

Instead of proving directly that we can assign an argument to each access ofH, we will prove this fact in a larger tree inv◦m(H), where m is big enough. The resultfollows then by restriction (compare Lemma B.1.7 and Corollary B.2.8 in Appendix B).By Lemma 4.2 the induced map between accesses is an orientation preserving coveringof degree n.In order to be able to assign an argument to each access we must provethat this map is expanding (compare Appendix B).

Take two consecutive periodic accessesAi = (vi, ℓi, ℓ′i) in H (i = 0, 1). The idea is to show that for some m big enough, theseaccesses are not consecutive in inv◦m(H).

As no new periodic vertices are added in theconstruction of inv◦m(H), we have no new periodic accesses and the conditions of LemmaB.1.7 are trivially satisfied; this will establish the result. We distinguish between v0 = v1and v0 ̸= v1.If v0 = v1 then ℓ0 ≺ℓ′0 ⪯ℓ1 ≺ℓ′1 ⪯ℓ0 at Ev0.

It is enough to find an m ≥0 suchthat inv◦m(H) has an access in the ‘branch’ Bv0,inv◦m(H)(ℓ′0). If there is a Julia vertexin Bv0,H(ℓ′0) this is obvious by Corollary 3.4.

If not, ℓ′0 has end points v0, ω where ω is aFatou point. Now the edge ℓ′0 corresponds to an argument in the coordinate φω at ω; as ωeventually maps to a critical point, we can find an argument θ ̸= φω(ℓ′0) which eventuallymaps to the same argument as φω(ℓ) under successive multiplication by degτ ◦i(ω) modulo 1(compare diagram (2) in §II.3.3).

It follows that for some m big enough, there is an ℓ′ ∈E′ωsuch that φω(ℓ′) = θ. The result then follows easily from Corollary 2.8.

(Alternatively, wecan use Corollary 2.9. )Now let v0, v1 be different periodic Julia points.

By Lemma II.3.7, for some m > 0 thereis a vertex v′ of inv◦m(H) in [v0, v1]T for otherwise H will not be expanding. If v′ is a Juliavertex we proceed as above.

Otherwise, we let (v′, ℓ, ℓ′) be the pseudoaccess (in inv◦m(H)at the Fatou vertex v′) between A0, A1 in the cyclic order. We take an argument θ betweenφω(ℓ) and φω(ℓ′) which eventually maps to the same argument as φω(ℓ) and proceed as inthe last paragraph.#4.4.

As every abstract Hubbard Tree H of degree n > 1 has a canonical extensionsatisfying the conditions of Proposition 4.3, we can associate to every access a coordinate30

compatible with the dynamics. Such map φH is called an external coordinate.

In practice,this will correspond to the argument of the external ray landing throughout this access.Now let θ 7→mn(θ) 7→. .

. 7→m◦kn (θ) = θ, be a periodic orbit under the standard n-foldmultiplication in R/Z.

The question is whether there is a canonical extension of H at whichaccesses corresponding to the arguments {θ, mn(θ), . .

. , m◦k−1n(θ)} are present.

For this wehave the following proposition.4.5 Proposition. Let H be a homogeneous abstract Hubbard Tree with at least oneJulia vertex.

For any election of external coordinate φH and periodic orbit θ 7→mn(θ) 7→. .

. 7→m◦kn (θ) = θ under n-fold multiplication in R/Z, there is a canonical extension of Hin which accesses corresponding to {θ, mn(θ), .

. .

, m◦k−1n(θ)} are present.Proof.Using Corollary 2.8 we may assume that the distance between two Fatouvertices in never equal to 1; and furthermore, whenever the distance between a Fatou and aJulia vertices is one, so is the distance between all their iterates. Also, because of Corollary2.9 we may assume without loss of generality that no Fatou vertex is an end.

We assumethat there are no accesses to which we can associate the referred periodic orbit and constructa canonical extension of this tree.Case 1. The easiest way to construct extensions with periodic orbits of period k iswhenever there is a Fatou periodic orbit of period dividing k. Suppose the total degree ofsuch critical cycle is d. In this case, for all arguments of period k under md we can includean edge which correspond in local coordinates to this argument and a periodic vertex (ifthey are not already present).

When this is done simultaneously at all Fatou vertices of thecycle we clearly get a new expanding Hubbard Tree. Clearly this construction is canonical.If the required accesses are present in this canonical extension, we are done; otherwise wehave to work harder.To continue the general case, first note that Corollary 3.4 guarantees that for m bigenough ν−m,H(v) = ν−∞,H(v) at every original vertex v ∈VH.

We will only keep track ofthe following information: the original tree H and all these accesses of inv◦m(H) at verticesv ∈VH in the original tree (we have ‘pruned’ the tree inv◦m(H)). In this case if ℓ∈Emvbut ℓ̸∈Ev (i.e, if the germ ℓat v in the tree inv◦m(H) is not present in H) we say that thetree inv◦m(H) was pruned at ℓ.Let {γ1, .

. .

, γα} be the arguments of all such accesses ordered counterclockwise. Work-ing if necessary in a canonical extension, we may further suppose that the Lebesgue measureof (γi, γi+1) is at most 1/n2k+2.

(In fact, we may work in an inverse inv◦l(H) with l bigenough thanks to the expansiveness of mn in R/Z.) It follows that (γi, γi+1) contains atmost one periodic orbit of period diving 2k in its closure.

In particular, each m◦in (θ) belongsto an interval (θ+i , θ−i ) with θ±istrictly preperiodic. It follows that the vertices v+θi, v−θi atthe respective accesses are not periodic.Suppose first that v+θ0 ̸= v−θ0.

By further subdividing the tree (for example by taking an31

extra k inverses and restricting to the vertices in the original underlying topological tree),we may suppose that for any edge ℓ, the iterated maps τ◦i|ℓare one to one (i = 1, . .

. k).Case 2.

Suppose [v+θ0, v−θ0] ⊂τ ◦k([v+θ0, v−θ0]). It follows from standard techniques forsubshifts of finite type that we can canonically extend the vertices of the tree so that itincludes an orbit of period k or k/2 with vm◦in (θ) ∈[v+θi, v−θi].

Because v±θi are strictly prepe-riodic, the expansive condition for the new set of vertices is trivially satisfied . Thereforeany access at vm◦in (θ) belonging to the set (θ+i , θ−i ) should have an associated argument ofperiod dividing 2k.

By construction this argument can only be m◦in (θ).Case 3. Suppose [v+θ0, v−θ0]∩τ ◦k([v+θ0, v−θ0]) = [v1, v2].

Then the vertices v1, v2 belong tothe interval [v+θ0, v−θ0]. Now, by hypothesis this last interval contains no vertex of Julia type(for otherwise after completing the accesses at such vertex, we will have that θ+0 and θ−0 arenot consecutive in the cyclic order) and at most one vertex w of Fatou type.

It follows thatv1 = w and that v2 equals either v+θ0 or v−θ0. In either case we get d(τ ◦k(v+θ0), τ◦k(v−θ0)) ≥3.However, by assumption this is impossible since d(τ◦k(v+θ0), τ◦k(w)) = d(τ ◦k(v−θ0), τ◦k(w)) =1 implies d(τ ◦k(v+θ0), τ◦k(v−θ0)) ≤2.Case 4.

Suppose [v+θ0, v−θ0] intersects τ ◦k[v+θ0, v−θ0] = [τ ◦k(v+θ0), τ◦k(v−θ0)] at an interiorvertex w ∈[v+θ0, v−θ0].It follows from the preliminary discussion in case 3 that w is aFatou vertex. This Fatou vertex should be periodic of period dividing k because otherwiseτ ◦k(w) ̸= w belongs to [τ◦k(v+θ0), τ◦k(v−θ0)] and therefore d(τ ◦k(v+θ0), τ◦k(v−θ0)) ≥3, whichcan be shown to be impossible as in case 3.Denote by ℓ±k the edges [w, τ◦k(v±θ0)] with local coordinates α±k at w, and by ℓ±0 theedges [w, v±θ0] with local coordinates α±0 .

Clearly (α+0 , α−0 ) ⊂(α+k , α−k ) because this is theonly ordering compatible with the order of the accesses. Denote by d the local degree of τ kat w.Claim.

md maps (α+0 , α−0 ) homeomorphicaly onto (α+k , α−k ).In fact, if this is not the case, in some inverse tree there is an edge ℓ′ = [w, v′] withcorresponding argument φw(ℓ′) ∈(α+0 , α−0 ) and with τ ◦k(ℓ′) = τ ◦k(ℓ+0 ).It follows thatafter completing the access at the vertex v′ there is an access with corresponding argumentβ ∈(θ+0 , θ−0 ) such that mn(β) = mn(θθ+0 ). But this implies that the interval (θ+0 , θ−0 ) hasLebesgue measure at least 1/nk, which is a contradiction.To finish the proof of case 4, we notice that the claim implies that md has a fixed pointinside the interval (α+0 , α−0 ).

Therefore we are in case 1.Case 5. Suppose the intervals [v+θ0, v−θ0] and [τ◦k(v+θ0), τ◦k(v−θ0)] have disjoint interiors.In this case we consider the subtree generated by the vertices v±0 and τ◦k(v±0 ) to notice thatthere is vertex v strictly contained in the interior of [τ◦k(v+θ0), τ◦k(v−θ0)].

Also there is anedge ℓat this vertex such that v±θ0 ∈B(ℓ) the branch at ℓ. In fact, this follows from the32

ordering of accesses. This implies in particular that for some inverse of the tree there is avertex v′ ∈[v+θ0, v−θ0] with τ ◦k(v′) = v. Also, we can find an edge ℓ′ at v′ which maps locallyto ℓunder τ k. If v′ is of Julia type, there are consecutive accesses (after completing theaccesses) at v′ with associated arguments θA and θB such that θ ∈(θA, θB) ⊂(θ+0 , θ−0 ).

If v′is of Fatou type, there is a Julia vertex v1 in the branch B(ℓ′) such that (after restriction tothe a tree which only includes this vertex in such branch) there are two consecutive accesseswith that property described above. In fact, these two properties follow immediately fromthe fact that accesses at v′ (respectively at v1) map to accesses at τ ◦k(v′) (respectively atτ ◦k(v1)), and that (θ+0 , θ−0 ) has Lebesgue measure at most 1/n2k+2.In either case we have reduced the problem to case 6.Case 6.

Suppose now that v+θ0 = v−θ0. After taking inverses and restricting if necessarywe may suppose that τ◦i(v±θ0) = v+θi for i = 0, .

. .

k −1. Thus, the accesses A+i and A−i withexternal arguments θ+i , θ−i share an edge ℓi.

As there is no further access with argument in(θ+i , θ−i ) it follows that some tree inv◦m(H) was “pruned” at ℓi. In this way, the requiredextension is achieved by adding the vertices vm◦in (θ) at the other end of ℓi.

Note that theextension is canonical because for any extension including the vertex v+θi, the vertex vm◦in (θ)should belong to the branch ℓi, and thus, according to Lemma II.3.13 these periodic verticesshould be ends.#4.6Corollary.EveryextensionofanabstractHubbardTreeiscanonical.Proof. Given any extension we assign to every periodic access its canonical argument(compare Proposition 4.3).

Then starting with the minimal tree we add all these periodicorbits according to Proposition 4.5. Finally, we take a finite number of inverses and restrictif necessary.#5.

From Hubbard Trees to Formal Critical Portraits.Using canonical extensions we will mimic the constructions of critical portraits fromthe first part of this work. For the main defenitions and results see Appendix A5.1 Extending the tree.Let H be an abstract Hubbard Tree of degree n > 1.We start with a canonical extension H′ of H as in Corollary 2.8; i.e, we require from thisextension that if ω is a Fatou point and ℓ∈Eω, then for the endpoints ω, v of ℓwe musthave that v is a Julia vertex, and dH′(τ ◦k(v), τ◦k(ω)) = 1 for all k ≥0.We fix local coordinates {φv}v∈V .

For any critical cycle we extend the tree by addingan edge and a vertex at every 0 argument (if they are not present). Next, for any Fatou33

vertex ω we proceed as follows. Inductively suppose that the 0 edge is present in the localcoordinate of τ(ω).

We insert a new vertex and edge (if they are not present) at everyargument of φ−1τ(ω)(0). Then we use Corollary 3.4 to guarantee that pseudoaccesses definedat such points are indeed accesses.

We call any extension satisfying the above conditionssupporting (compare §I.2).Let ω be a Fatou vertex, an access (v, ℓ′, ℓ) is said to support ω if ℓhas endpoints ω, vand dH(τ ◦k(ω), τ◦k(v)) = 1 for all k ≥0. Clearly τ(v, ℓ′, ℓ) = (τ(v), τv(ℓ′), τv(ℓ)) supportsτ(ω).

An access (v, ℓ′, ℓ) which supports the Fatou critical point ω will be denoted by D(ω, ℓ)5.2 Constructing marked accesses.Let H be a supporting abstract HubbardTree. Using Corollary 3.4 we pick an inverse inv◦m(H) such that at every v ∈V we haveνH,−m(v) = νH,−∞(v).

From this it is easy to chose hierarchic accesses as in §I.2:For each critical vertex ω ∈Ω(H) setΛω = {ℓ∈Eω : δ(ω)φv(ℓ) = 0}(in this case the hierarchic selection is reflected in the choice of a 0 argument in the localcoordinate). Let Ω(F) = {ωF1 , .

. .

, ωFl } be the set of Fatou critical vertices, and Ω(J ) ={ωJ1 , . .

. , ωJk } the set of Julia critical vertices.

For each ω ∈Ω(F) we construct δ(ω) markedsupporting accesses to ω in the following way. Take ℓ∈Λω with end points vℓ, ω; then thereis a supporting access to ω at vℓof the form D(ω, ℓ) = (vℓ, ℓ′, ℓ).

The set of such accessesfor all possible ℓ∈Λω is by definition Fω.For each ω ∈Ω(J ) we construct δ(ω) marked accesses in the following way.Takeℓ∈Λω, then there is an accesses at ω of the form E(ω, ℓ) = (ω, ℓ, ℓ′). The set of suchaccesses for all possible ℓ∈Λω is by definition Jω.Note the slight difference in the construction, at a Julia critical vertex v, the markedaccesses are at v. While for Fatou critical vertices the accesses are taken at the other endof each edge.In this way we have constructed two familiesF= {Fω1, .

. .

, Fωl}J= {Jω1, . .

. , Jωk}of accesses.

As these accesses correspond in the external coordinate φH to arguments, wewill not distinguish between the accesses and their corresponding argument. In this way wehave the following (see §I.3).5.3 Proposition.

The marking (F, J ) is a formal critical portrait.Proof. This follows directly from the construction.#34

There are several trivial consequences of this construction that we want to point out.To simplify notation, the vertex at which an access C is defined will be denoted by vC. Theproof in all cases is the same: by removing the edge ℓwe are left with two connected pieces.5.4 Lemma.

Let ω be a Fatou critical vertex. If vC ∈BH,ω(ℓ), then for all ℓ′ ∈Λω−{ℓ}we have D(ω, ℓ′) ≺C ⪯D(ω, ℓ).#5.5Lemma.LetωbeaJuliacriticalvertex,andCanaccessatvC∈BH,ω(ℓ) −{ω}.ThenforanyaccessesA, A′atωwehaveeitherA ≺C ≺A′ or A′ ≺C ≺A.#5.6 Lemma.

Suppose ω is a Fatou critical vertex and let ℓ̸∈Λω. If C an accessat vC ∈BH,ω(ℓ), then for any ℓ′, ℓ′′ ∈Λω we have either D(ω, ℓ′) ≺C ≺D(ω, ℓ′′) orD(ω, ℓ′′) ≺C ≺D(ω, ℓ′).#6.

From Hubbard Trees to Admissible Critical Portraits.In this section we prove that the formal critical portrait constructed above is alsoadmissible. For this we must verify conditions (c.6), (c.7) in §A.2.7.

We first verify condition(c.6). The verification of condition (c.7), will also show that any polynomial with criticalmarking (F, J ) has Hubbard Tree equivalent to this starting one.

In this way the mainTheorem A will follow.6.1 Proposition. The formal critical portrait (F, J ) is an admissible critical portrait.Proof.

This follows from Corollaries 6.4 and 6.9 below.#6.2 Lemma. Let Ai, Bi be accesses at vi for i = 1, 2 with v1 ̸= v2.

Then {A1, B1},and {A2, B2} are unlinked.Proof. This follows from the fact that {A2, B2} are defined in the same connectedcomponent of T −{v1}.#6.3 Lemma.

Let A, A′ be periodic accesses. If either S+(A) = S+(A′) or S−(A) =S−(A′), then vA = vA′.Proof.

By contradiction suppose vA ̸= vA′. We distinguish two cases.35

Suppose τ ◦k|[vA,vA′]T is injective for all k ≥1. In this case there is a periodic Fatouvertex v ∈[vA, vA′]T , because otherwise the tree will not be expanding.

Let d > 1 be thedegree of the critical cycle v0 = v →v1 . .

. →vn = v0.

There are exactly two differentedges ℓ, ℓ′ ∈Ev contained in [vA, vA′]T . The dynamics of these edges must be periodicby Lemma II.3.7.

We write φv(ℓ), φv(ℓ′) in base d expansion. As they are not equal byhypothesis, we may suppose that the first coefficient in the expansions are different.

As dis the product of the degrees of the vertices in the cycle, we may suppose then that whenmultiplying by δ(v0) they have different integer part. But in this way by Lemma 5.6 wewill have π0(S+(A)) ̸= π0(S+(A′)).

(In fact, for ǫ > 0 small enough, the arguments φH(A)and φH(A′) belong to different connected components of R/Z −{φH(D(v, ℓ)) : ℓ∈Λv} =R/Z−Fv.) But implies that S+(A) ̸= S+(A′).

If we consider instead of φv the ‘coordinate’1 −φv the same reasoning give us S−(A) ̸= S−(A′).Suppose now that τ|[vA,vA′]T is not locally one to one near ω. If ω is a Julia criticalvertex the result follows from Lemma 5.5.

If ω is a Fatou critical vertex, by Lemma 5.6 wealways have π0(S−(A)) ̸= π0(S−(A′)) and thus S−(A) ̸= S−(A′).If neither A nor A′ support ω, again by Lemma 5.6 π0(S+(A)) ̸= π0(S+(A′)). Westart though by assuming that A is a marked access associated with ω.

By Hypothesisthere is a preperiodic marked access C ∈Fω (and therefore such that τ(C) = τ(A)) withvC ∈[vA, vA′]T .Thus τ◦k|[vC,vA′]T eventually maps into [vA, vA′]T .It follows there isa point ω′ ∈[vC, vA′]T that eventually maps to ω. Working if necessary in a canonicalextension inv◦k(H) we may assume without loss of generality that ω′ ∈V .

But then byLemma II.3.7 for some i ≥k, τ ◦i|[ω,ω′]T is not locally one to one near some point ω′′. Ifi is minimal, neither of the periodic accesses τ ◦i(A) = τ ◦i(C) nor τ ◦i(A′) can support thecritical point τ ◦i−1(ω′′) if it is of Fatou type.

It follows from the previous reasoning thatS+(τ ◦i−1(A)) ̸= S+(τ ◦i−1(A′)), and therefore S+(A) ̸= S+(A′).#6.4 Corollary. The formal critical portrait (F, J ) satisfies condition (c.6).Proof.

Let A be a periodic marked access. Suppose there is a periodic argument λsuch that S+(λ) = S+(A).

By Proposition 4.5 we can assume that there is an accessescorresponding to λ. By Lemma 6.3 this access is supported at vA. By Lemma 5.4 thisaccess can only be A.#6.5 Lemma.

Let vA = vA′ be a non critical Julia vertex. Then A and A′ have thesame left address, i.e, π0(S−(A)) = π0(S−(A′)).Proof.

If E, E′ are marked accesses associated to the same Julia critical vertex, Lemma6.2 implies that {A, A′},{E, E′} are unlinked.If D, D′ are marked accesses associated to the same Fatou critical vertex, we distinguishif vA equals vD or not. If vA ̸= vD, vD′ then clearly {A, A′},{D, D′} are unlinked becausethe regulated path [vD, vD′]T does not contain vA.

If vA = vD then by Lemma 5.5 D′ ≺A ≺A′ ⪯D.36

All these facts together mean by definition that the accesses A and A′ have the sameleft address, i.e, π0(S−(A)) = π0(S−(A′)).#6.6 Lemma. Let B be an access at a Julia critical vertex v. Then there is a markedaccess E at v, such that π0(S−(E)) = π0(S−(B)).Proof.TakeconsecutiveE,E′markedaccessesatv,suchthatA′ ≺E ⪯A.

Using Lemma 6.2 and the same reasoning as in Lemma 6.5 we get π0(S−(A)) =π0(S−(E)).#6.7 Corollary.Suppose π0(S−(A)) = π0(S−(A′)).Then vA = vA′ if and only ifvτ(A) = vτ(A′).Proof. One direction is obvious.

On the other hand, we may assume that vτ(A) has ninverses in the tree counting multiplicity. As there are only n possible choices of addresses,the result follows combining Lemmas 6.3, 6.5, 6.6.#6.8 Proposition.

vA = vA′ if and only if S−(A) ∼l S−(A′).Proof. First suppose S−(A) ∼l S−(A′).

It is enough to prove that if S−(A) ≈S−(A′)then vA = vA′. If S−(A) = S−(A′) this follows from Lemma 6.3 and Corollary 6.7.

In theother case the result follows from this fact, Lemma 6.6 and again Corollary 6.7.Suppose now vA = vA′. Let m ≥0 be the smallest integer such that τ ◦m(vA) doesnot contain in its forward orbit a critical vertex.

The proof will be in induction in m. Form = 0 this is Lemma 6.5. Suppose now that the result holds for m−1.

This implies that allaccesses at τ(vA) have equivalent symbol sequences. If v is not critical we use again Lemma6.5.

If v is critical we use Lemma 6.6.#6.9 Corollary. The formal critical portrait (F, J ) satisfies condition (c.7).#37

7. Proof of the Theorem A.The admissible critical portrait (F, J ) determines a unique (up to affine conjugation)polynomial P with marking (P, F, J ) by Theorem A.2.9.

By Propositions 6.8 and A.2.12its Hubbard Tree is the starting one. The angle function at Fatou vertices are the startingones because of Proposition 2.7, and Corollaries 2.8 and B.2.5#38

Appendix ACritical Portraits.We follow closely the exposition in [P1] about critical portraits for postcritically finitepolynomials. Proofs of all statements and further details can be found in said work.§A.1 Construction of Critically Marked Polynomials.A.1.1 Supporting arguments.

Let P be a PCF Polynomial. Given a Fatou com-ponent U and a point p ∈∂U, there are only a finite number of external rays Rθ1, .

. .

, Rθklanding at p. These rays divide the plane in k regions. We order the arguments of these raysin counterclockwise cyclic order {θ1, .

. .

, θk}, so that U belongs to the region determinedby Rθ1 and Rθ2 (θ1 = θ2 if a single ray lands at p). The argument θ1 (respectively the rayRθ1) is by definition the (left) supporting argument (respectively the (left) supporting ray) ofthe Fatou component U.

In a completely analogous way we can define right supporting rays.Note that an argument supports at most one Fatou component. Furthermore, by definition,given a Fatou component U, at every boundary point p lands a supporting ray for U.Definition.

Given an external ray Rθ supporting the Fatou component U(z) withcenter z, we extend Rθ by joining its landing point with z by an internal ray, and call thisset an extended ray ˆRθ with argument θ.Given a postcritically finite polynomial P, which we assume to be monic and centered,we associate to every critical point a finite subset of Q/Z and construct a critically markedpolynomial (P, F = {F1, . .

. , FnF }, J = {J1, .

. .

, JnJ }).Here Fk would be the set ofarguments associated with the critical point zFk in the Fatou set, and Jk would be theset associated with the critical point zJk in the Julia set. We remark that given a polynomialits critical marking is not necessarily unique.

Also note that one of these two families maybe empty if there are no critical points in the Fatou or Julia sets. In the following definitionwe will always work with left supporting rays.

We remark that we could equally well workwith the right analogue, but there must be the same choice throughout. Also, multiplicationby d modulo 1 in R/Z will be denoted by md.A.1.2 Construction of Fk.

First we consider the case in which a given Fatou criticalpoint z = zFk is periodic. Let z = zFk 7→P(z) 7→... 7→P ◦n(z) = z be a critical cycle of39

period n and degree Dz > 1 (by definition the degree of a cycle is the product of the localdegree of all elements in said cycle). We construct the associated set Fk for every criticalpoint in the cycle simultaneously.

Denote by dz be the local degree of P at z. We pick anyperiodic point pz ∈∂U(z) of period dividing n (which is not critical, because is periodicand belongs to the Julia set J(P)) and consider the supporting ray Rθ for this componentU(z) at pz.

Note that this choice naturally determines a periodic supporting ray for everyFatou component in the cycle. The period of this ray is exactly n. Given this periodicsupporting ray Rθ, we consider the dz supporting rays for this same component U(z) thatare inverse images of P(Rθ) = Rmd(θ).

The set of arguments of these rays is defined to beFk. Keeping in mind that a preferred periodic supporting ray has been already chosen, werepeat the same construction for all critical points in this cycle.

Note that as the cycle hascritical degree Dz, we can produce Dz −1 different possible choices for Fk. If Fk is theset associated with the periodic critical point zk, there is only one periodic argument in Fk(namely θ as above), we call this angle the preferred supporting argument associated withzFk .

Note that by definition, the period of zFk equals the period of the associated preferredperiodic argument.Otherwise, if z = zFk of degree dz > 1, is a non periodic critical point in the Fatou setF(P), there exists a minimal n > 0 for which w = P ◦n(z) is critical. If w has associateda preferred supporting ray Rθ (at the beginning only periodic critical points do), then inP −n(Rθ) there are exactly dz rays that support this Fatou component U(z).

The set ofarguments of these rays is defined to be Fk. We pick any of those and call it the preferredsupporting argument associated with z.

We continue this process for all Fatou critical points.A.1.3 Construction of Jk. Given z = zJk (a critical point in J(P)) of degree dk > 1,we distinguish two cases.

If the forward orbit of z contains no other critical point, we havethat for some θ (usually non unique) Rθ lands at P(z). Now P −1(Rθ) consists of d differentrays, among them exactly dk land at z. Define Jk as the set of arguments of these rays,and choose a preferred ray.

Otherwise, z will map in n ≥1 iterations to a critical point,which we assume to have associated a preferred ray Rθ. In the nth inverse image P −n(Rθ)of this preferred ray, there are dk rays which land at z.

The set of arguments of these raysis defined to be Jk. Again we pick one of those to be preferred, and continue until everycritical point has an associated set.The critical marking itself gives information about how many iterates are needed for agiven critical point to become periodic.

For example, by construction we have the followinglemma.A.1.4 Lemma. Let γ be a preferred supporting argument in Fk (respectively in Jk).Then the multiple m◦nd (γ) (with n ≥1) is periodic but m◦n−1d(γ) is not if and only if zFk(respectively zJk ) falls in exactly n iterations into a periodic orbit.Remark.

Note that the construction of associated sets was done in several steps. Wefirst complete the choice for all critical cycles, and then proceed backwards.

In both theFatou and Julia set cases we will have to make decisions at several stages of the construction.40

Such decisions will affect the choice of the marking for all critical points found in thebackward orbit of these starting ones. Each time that this kind of construction is made, wewill informally say that it is a hierarchic selection.§A.2 The Combinatorics of Critically Marked Polynomials.In order to analyze which conditions the families (F, J ) satisfy, it is convenient tointroduce some combinatorial notation.A.2.1 Definitions.

We say that a subset Λ ⊂R/Z is a (d-)preargument set if md(Λ)is a singleton. For technical reasons we assume always that Λ contains at least two elements.If all elements of Λ are rational, we say that Λ is a rational preargument set.

It follows byconstruction that whenever (P, F, J ) is a marked polynomial, all the sets Jk, and Fl arerational d-preargument sets.Consider now a family Λ = {Λ1, . .

. , Λn} of finite subsets of the unit circle R/Z.

Thefamily Λ determines the family union set Λ∪= S Λi. We say that any λ ∈Λ∪is an elementof the family Λ.

Furthermore, we can say that it is a periodic or preperiodic element of thefamily if it is so with respect to md. The set of all periodic elements in the family unionwill be denoted by Λ∪per.A.2.2 Hierarchic Families.

We say that a family Λ is hierarchic if for any elementsin the family λ, λ′ ∈Λ∪, whenever m◦id (λ), m◦jd (λ′) ∈Λk for some i, j > 0 then m◦id (λ) =m◦jd (λ′). (This is useful if we think of a dynamically preferred element in each Λk).A.2.3 Linkage Relations.

We will say that two subsets T and T ′ of the circle R/Zare unlinked if they are contained in disjoint connected subsets of R/Z, or equivalently, if T ′is contained in just one connected component of the complement R/Z −T . (In particularT and T ′ must be disjoint.) If we identify R/Z with the boundary of the unit disk, anequivalent condition would be that the convex closures of these sets are pairwise disjoint.If T and T ′ are not unlinked then either T ∩T ′ ̸= ∅or there are elements θ1, θ2 ∈T andθ′1, θ′2 ∈T ′ such that the cyclic order can be written θ1, θ′1, θ2, θ′2, θ1.

In this second casewe say that T and T ′ are linked. More generally, a family Λ = {Λ1, .

. .

, Λn} is an unlinkedfamily if Λ1, . .

. , Λn are pairwise unlinked.

Alternatively each Λi is completely contained ina component of R/Z −Λj for all j ̸= i.The preceding definition has its motivation in the description of the dynamics of externalrays in polynomial maps. Suppose the external rays Rθi, Rψi land at zi for i = 1, 2.

If z1 ̸= z2then the sets {θ1, ψ1}, {θ2, ψ2} are unlinked, for otherwise the rays will cross each other.The same argument applies if we consider rays supporting different Fatou components. Butif we analyze linkage relations arising from rays supporting a Fatou component and rays41

that land at some point, we may get minor problems. Anyway, it is easy to see that evenin this case the associated sets of arguments will be ‘almost’ unlinked.

(Compare condition(c.2) and as well as Proposition A.2.8 below. )A.2.4 Weak linkage relations.Consider families F = {F1, .

. .

, Fn} and J ={J1, . .

. , Jm}; we say that J is weakly unlinked to F in the right if we can chose arbitrarilysmallǫ>0sothatthefamily{F1, .

. .

, Fn,J1 −ǫ, . .

. , Jm −ǫ} is unlinked.

(Here Λ −ǫ = {λ −ǫ (mod 1) : λ ∈Λ}. )In particu-lar each family should be unlinked.

Note that the definition allows empty families. Tosimplify notation we will simply say that “F and J −are unlinked”.A.2.5 Formal Critical Portraits.

Let F = {F1, . .

. , Fn} and J = {J1, .

. .

, Jm} betwo families of rational (d-)prearguments. We say that the pair (F, J ) is a degree d formalcritical portrait if the following conditions are satisfied.

(c.1) d −1 = P(#(Fk) −1) + P(#(Jl) −1)(c.2) “F and J −are unlinked”. (c.3) Each family is hierarchic.

(c.4) Given γ ∈F∪, there is an i > 0 such that m◦id (γ) ∈F∪per. (c.5) No θ ∈J ∪is periodic.This set of conditions represent the simplest conditions satisfied by the critical markingof a postcritically finite polynomial.

Condition (c.1) says that we have chosen the right num-ber of arguments. Condition (c.2) means that the rays and extended rays determine sectorswhich do not cross each other, and that F was constructed from arguments of left supportingrays.

This reflects our decision to chose the supporting arguments as the rightmost possibleargument of an external ray. Condition (c.3) reflects our choice of preferred rays.

Condition(c.4) indicates that arguments in F are related to Fatou critical points. Condition (c.5)indicates that arguments in J are related to Julia set critical points.

Unfortunely there areformal critical portraits which do not correspond to a postcritically finite polynomial. Inorder to state necessary and sufficient conditions we need to study the dynamically definedpartitions of the unit circle determined by these elements.A.2.6.

Given two families F, J as above, we form a partition P = {L1, . .

. , Ld} ofthe unit circle minus a finite number of points R/Z −F∪−J ∪, in the following way.

Weconsider two points t, t′ ∈R/Z −F∪−J ∪. By definition, t, t′ are unlink equivalent if theybelong to the same connected component of R/Z −Fi and R/Z −Jj, for all possible i, j.Let L1, .

. .

, Ld be the resulting unlink equivalence classes with union R/Z −F∪−J ∪. It iseasy to check that each Lp is a finite union of open intervals with total length 1/d.42

Each element Li ∈P of the partition is a finite union Li = ∪(xj, yj) of open connectedintervals.WedefinethesetsL+i=∪[xj, yj)andL−i=∪(xj, yj].ItiseasytoseethatbothP+={L+1 , . .

. , L+d }andP−= {L−1 , .

. .

, L−d } are partitions of the unit circle. As every θ ∈R/Z belongs to exactlyone set L+k , we define its right address A+(θ) = Lk.

In an analogous way we define the left ad-dress A−(θ) of θ. We associate to every argument θ ∈R/Z a right symbol sequence S+(θ) =(A+(θ), A+(md(θ)), .

. .) and a left symbol sequence S−(θ) = (A−(θ), A−(md(θ)), .

. .

). Notethat for all but a countable number of arguments θ ∈R/Z (namely the arguments present inthe families and their iterated inverses), the left S−(θ) and the right S+(θ) symbol sequencesagree.

By S(θ) will be meant either (left or right) symbol sequence.A.2.7 Admissible Critical Portraits. Let F = {F1, .

. .

, Fn}, J = {J1, . .

. , Jm}be two families of rational (d-)prearguments.

We say that the pair (F, J ) is a degree dadmissible critical portrait if (F, J ) is a degree d formal critical portrait and the followingtwo extra conditions are satisfied. (c.6) Let γ ∈F∪per and λ ∈R/Z, then λ = γ if and only if S+(γ) = S+(λ).

(c.7) Let θ ∈Jl and θ′ ∈Jk. If for some i, S−(m◦id (θ)) = S−(θ′), then m◦id (θ) ∈Jk.A.2.8 Proposition.

If (P, F, J ) is a critically marked polynomial, then (F, J ) is anadmissible critical portrait.Condition (c.6) indicates that arguments in Fl must support Fatou components. Con-dition (c.7) indicates that different elements in the family J are associated with differentcritical points.

Now we can state the main result for critically marked polynomials as follows.A.2.9 Theorem. Let (F, J ) be a degree d admissible critical portrait.

Then there isa unique monic centered postcritically finite polynomial P, with critical marking (P, F, J ).Now we should ask if conditions (c.1)-(c.7) represent a finite amount of information tobe checked. This question is answered in a positive way by the following proposition.A.2.10 Lemma.

Suppose θ and θ′ have the same periodic (left or right) symbol se-quence. Then θ and θ′ are both periodic and of the same period.A.2.11.

The next question that we ask is what kind of information about the Juliaset can be gained by looking carefully into the combinatorics.For example, if can wedetermine if two rays land at the same point by only looking at their arguments. In fact,left symbol sequences contain all the information necessary to determine whether two raysland at the same point or not.

This is done as follows. Suppose Ji = {θ1, ..., θk} ∈J withcorresponding left symbol sequences S−(θ1), .

. .

, S−(θk). As we expect the rays with those43

arguments to land at the same critical point, we declare them (i-)equivalent; i.e, we writeS−(θα) ≡i S−(θβ). Then we set θ ≈θ′ either if S−(θ) = S−(θ′) or there is an n ≥0 suchthat A−(m◦jd (θ)) = A−(m◦jd (θ′)) for all j < n and S−(m◦nd (θ)) ≡i S−(m◦nd (θ′)) for some i.This relation ≈is not necessarily an equivalence relation, because transitivity may fail.

Tomake this into an equivalence relation we say that θ ∼l θ′ if and only if there are argumentsλ0 = θ, λ1, . .

. , λm = θ′, such that λ0 ≈.

. .

≈λm. The importance of this equivalencerelation is shown by the following proposition.A.2.12 Proposition.

Let (P, F, J ) be a critically marked polynomial. Then Rθ andRθ′ land at the same point if and only if θ ∼l θ′.A.2.13 Corollary.

The symbol sequence S−(θ) is a periodic sequence of period m ifand only if the landing point of the ray Rθ has period m.#Appendix BFinite Cyclic Expanding Maps.1. Expanding Maps.We consider a finite cyclic set X, and a degree n ≥2 orientation preserving mapf : X 7→f(X) ⊂X.

We will study under which conditions we can assign an argument φ(p)to every point p ∈X such that the induced map becomes multiplication by n.1.1. Let k ≥1 and n ≥2.

Consider a finite cyclicly ordered set X = {p1, . .

. , pkn}with kn elements.

The cyclic order can be realized as a successor function SucX(pi) = pi+1with the convention pkn = p0. Given Y ⊂X there is an induced order in Y , and therefore asuccessor function SucY : Y 7→Y .

We consider a degree n ≥2 orientation preserving map44

f : X 7→f(X) ⊂X. By this we mean a function f with the property that f(pi) = f(pj) ifand only if i ≡j (mod k); and such that f(SucX(p)) = Sucf(X)(f(p)).

It follows that f isan nth-fold cover of its image. Note that because f is a degree n cover and order preserving,for every p ∈X, the restriction of f to the set {p, SucX(p), .

. .

, Suc◦k−1X(p)} is one to oneand onto f(X).Given a cyclicly ordered set X as above, we define the ordered distance dX(p1, p2)between two points p1, p2 ∈X, as the minimal m for which p2 = Suc◦m(p1). Thus, theordered distance between two points is always less than kn.

It follows easily that f(p1) =f(p2) if and only if dX(p1, p2) is a multiple of k.Given three points p1, p2, p3 and numbers 0 ≤m ≤m′ < kn, with m = dX(p1, p2),m′ = dX(p1, p3), we write p1 ≤p2 ≤p3. If in addition m < m′ we write p1 ≤p2 < p3.1.2Lemma.Supposep1≤p2≤p3

Completely trivial.#1.3 Remark. Even if we are considering two orders (one in X and that induced inf(X)), we will only be considering the ordered distance of X.

In other words if p1, p2 ∈f(X),the ordered distance dX(p1, p2) is always measured in X.1.4 Definition. We say that f : X →X as above is expanding, if given p1, p2 periodic(⋆) there exists l ≥0 such that dX(f ◦l(p1), f ◦l(p2)) ̸= 1.In other words, if two periodic points are consecutive, the distance between them even-tually increases.

From the facts that dX(p1, p2) < k implies f(p1) ̸= f(p2), and every pointis eventually periodic, we can easily deduce that for an expanding map, condition (⋆) is alsosatisfied for every pair of different points.1.5 Given a finite cyclicly ordered set X and a degree n ≥2 orientation preserving mapf, we say that f : X →X can be angled if there is an order preserving embedding φ : X 7→R/Z, such that nφ(p) ≡φ(f(p))(mod 1). Of course, an angled function is expanding.1.6 Remark.

If we reverse the order in all the definitions above (i.e, if we replace thesuccessor function by a predecessor function PreX), all the definitions above make sense.In particular if φS,φP are the angle functions for these two orders then clearly φS + φP ≡1.1.7 Proposition. Let X be a finite cyclic set and f an orientation preserving degreen ≥2 map.

Then f : X →X is angled if and only if is expanding.45

Proof. Being angled implies being expanding as remarked above.

We prove the con-verse in several steps.Step 1: We can assume without loss of generality that f has a fixed point. In fact,if there is no fixed point, then f(X) has at least two elements.We define a functiong : X →{1, .

. .

, kn −1} by the formula g(x) = dX(x, f(x)). It follows easily from Lemma1.2 that whenever i ≡j (mod k) then g(xi) ≡g(xj) + dX(xi, xj) (mod kn).

Thereforefor y ∈f(X), there is a unique xi ∈f −1(y) for which k(n −1) < g(xi) < kn (in fact,g(xi) = k(n −1) would imply that xi+k is a fixed point). Let d be the maximum of g.Among all x with g(x) = d take one for which g(SucX(x)) < d. It follows easily that thecyclic order can be written asf(x) < x < SucX(x) < f(SucX(x)) = Sucf(X)(f(x)) < f(x).To simplify notation, we rewrite X as {p0 = x, p1, .

. .

, pkn−1}. We insert a new point qibetween every pair pki and pki+1.

All of this new points will be mapped to q0. In this way,we have a degree n ≥2 orientation preserving map which is an extension of the original one.We must verify that this map is expanding.

The only new periodic point included is q0.The expanding property is obviously verifies if SucX(q0) is periodic: if dX(f(q0), f(SucX(q0))) =1 then SucX(q0) is a fixed point, in contradiction to what was assumed. If PreX(q0) is pe-riodic the result follows analogously.Step 2: We assign an argument to each point in X as follows.

Let q0 < q1 < qn−1 < q0be all points which map to the fixed point q0.We assign to qi the argument i/n fori = 0, . .

. , n −1.

For an arbitrary point x ∈X, we dynamically find its numerical expansionin base n.Step 3: The assignment is order preserving. Because the function is n to one orderpreserving, we may introduce inverse iterates of the fixed point.

Thus, we may assume thatgiven m there are in the cyclic order different values {q0 = q, . .

. , qmn−1} with the propertythat f ◦m(qi) = q0.

Taking m big enough the result follows.Step 4: Different points are assigned different arguments. Consider a set {x1, .

. .

, xl} ofmaximal cardinality to which equal periodic base n expansion is associated. Clearly all xi areperiodic.

Furthermore, if l > 1 we have for all m ≥0 dX(f ◦m(x1), f ◦m(x2)) = 1 because ofmaximality. But this contradicts the expanding condition.

There is a case in which this argu-ment does not apply. Suppose that in applying step 2, there is an argument to which the deci-malexpansion0.n−1, n−1, .

. .isassigned.Inthiscasewereversetheorder,andapplythesameargumenttoderiveacontradiction.#2.

Finding the Coordinates.Consider an integer n > 1, and denote by mn multiplication by n modulo 1. From thedynamical point of view the election of 0 as the origin is arbitrary in the sense that any46

dynamically property present at a point x ∈R/Q, is also present at x + j/(n −1). In thisway, with the knowledge of the dynamical behavior of a point x, the natural question is notwhich is the value of x, but that of mn−1(x).2.1 For n > 1 define δn : R/Z →{0, .

. .

, n −1} byδn(x) = iifmn−1(x) ∈[ in, i + 1n).In other words, if we take mn−1(x), we define δn(x) as the integer part of n(mn−1(x)).2.2 Remark. It follows that δn(x) is the number of inverses of mn(x) (other thanx) in the cyclicly counterclockwise oriented interval (x, md(x)) (if x is fixed this interval isinterpreted to be empty).

To see this, we rewrite mn−1(x) as mn(x) −x (mod n). In thisway, δn(x) counts the number of intervals of size 1/n to be found in (x, md(x)).

The claimfollows easily.2.3 Example. Consider with n = 3 the point x = 1/5.

We have m2(x) = 2/5 and1 ≤3(2/5) < 2; so by definition δ3(1/5) = 1. Note also that δ3(1/5 + 1/2) = 1, which is nota surprise because m2(x) = m2(x + 1/2) for all x ∈R/Z.2.4 Lemma.

Let n > 1 and x ∈R/Z, thenmn−1(x) = 1n∞Xi=0δn(m◦in (x))ni.Proof. We successively subdivide the interval Ij = [jn−1, j+1n−1) in n semiopen intervals.This determines a parametrization of the interval Ij by symbol sequences in the symbolspace {0, .

. ., n−1} (not allowing any symbol sequence with tail (n−1, n−1, .

. .

)). A pointx ∈Ij has symbol sequence S0, S1, .

. .

if and only if x =jn−1 +1(n−1)nPi=0Sini . Therefore,mn−1(x) = 1nP∞i=0Sini (and all reference to the initial interval Ij is lost).

The result followsas Si = δn(m◦in (x)) by construction.#2.5 Coordinates for expanding maps. We return to the case described in §1.

Itfollows by definition of covering map and Remark 2.2 that δn(x) equals the integer partof dX(x,f(x))k. Thus, according to Lemma 2.4, mn−1(x) is independent of the coordinateassigned in Proposition 1.7. Furthermore, we have proved the following.2.6 Theorem.

Let X be a finite cyclic ordered set and f : X 7→X be a degree norientation preserving expanding map. Then f can angled in exactly n −1 ways.47

2.7 Example. (Compare Figure B.1.) Let X be the cyclic set shown in Figure B.1.

(The notation is justified by the dynamics.) We consider a map f : X →X for whichf(A) = f(A′) = f(A′′) = Bf(B) = f(B′) = f(B′′) = Cf(C) = f(C′) = f(C′′) = Df(D) = f(D′) = f(D′′) = A.The unique periodic orbit is given by A 7→B 7→C 7→D 7→A.

This map is clearlyexpanding. According to Remark 2.5, we have that δ(A) = 1, δ(B) = 0, δ(C) = 1, andδ(D) = 2.

Using Lemma 2.4, we can easily find the base 3 expansions of m2(A) = 0.1012.It follows that m2(A) = 2/5, and therefore A takes value either 1/5 or 7/10.A"BCDA’C"DB"AC’D"B’Figure B.12.8 Corollary.Let x be periodic under mn, and denote by O(x) its orbit.Thenmn−1(x) is uniquely determined by the cyclic order of m−1n (O(x)).Proof. This follows directly from Remark 2.2 and Lemma 2.4.

(Compare also Example2.7).#48

References. [BFH] B. Bielefeld, Y.Fisher, J. Hubbard, The Classification of Critically PreperiodicPolynomials as Dynamical Systems; Journal AMS 5(1992)pp.

721-762. [DH1] A.Douady and J.Hubbard, ´Etude dynamique des polynˆomes complexes, part I;Publ Math.

Orsay 1984-1985. [DH2] A.Douady and J.Hubbard, A proof of Thurston’s Topological Characterizationof Rational Maps; Preprint, Institute Mittag-Leffler 1984.

[F] Y.Fisher, Thesis; Cornell University, 1989. [GM] L.Goldberg and J.Milnor, Fixed Point Portraits; Ann.

scient. ´Ec.

Norm. Sup.,4e s´erie, t. 26, 1993, pp 51-98[L] P.Lavaurs, These; Universit´e de Paris-Sud Centre D’Orsay; 1989.

[M] J.Milnor, Dynamics in one complex variable:Introductory Lectures; Preprint#1990/5 IMS SUNY@StonyBrook. [P1] A.Poirier, On Postcritically Finite Polynomials, Part One:Critical Portraits;Preprint #1993/5 IMS SUNY@StonyBrook.

[P2]A.Poirier,Thesis;OnPostcriticallyFinitePolynomials;SUNY@StonyBrook 1993.49

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